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UBC Theses and Dissertations

Temperature and dislocation stress field models of the LEC growth of gallium arsenide 1986

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TEMPERATURE AND DISLOCATION STRESS FIELD MODELS OF THE LEC GROWTH OF GALLIUM ARSENIDE by CARLOS ENRIQUE SCHVEZOV L i e . en F i s i c a , U n i v e r s i d a d N a c i o n a l de R o s a r i o , A r g e n t i n a , 1975 M.A.Sc. i n M e t a l l u r g y , U n i v e r s i t y of B r i t i s h C o l u m b i a , Canada, 1983 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (De p a r t m e n t of M e t a l l u r g i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g to t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1986 © C a r l o s E n r i q u e S c h v e z o v , 1986 S In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) 11 ABSTRACT The t e m p e r a t u r e f i e l d s and r e s u l t i n g s t r e s s f i e l d s have been c a l c u l a t e d f o r a g r o w i n g GaAs c r y s t a l p r o d u c e d by t h e LEC p r o c e s s . The c a l c u l a t i o n s a r e b a s e d i n a f i n i t e e l e m e n t n u m e r i c a l t h e r m o e l a s t i c s t r e s s a n a l y s i s . The c a l c u l a t e d t e m p e r a t u r e f i e l d s have been compared t o r e p o r t e d e x p e r i m e n t a l measurements w i t h good a g r e e m e n t . The s t r e s s f i e l d s have been used t o c a l c u l a t e t h e r e s o l v e d s h e a r s t r e s s e s , i n t h e g r o w i n g c r y s t a l , from w h i c h t h e d i s l o c a t i o n d e n s i t y and d i s t r i b u t i o n were d e t e r m i n e d . U s i n g t h e model t h e e f f e c t s of a range of growth and e n v i r o n m e n t a l p a r a m e t e r s on the d i s l o c a t i o n d e n s i t y and d i s t r i b u t i o n were d e t e r m i n e d . T h e s e s p a r a m e t e r s i n c l u d e c r y s t a l l e n g t h , c r y s t a l d i a m e t e r , cone t a p e r , b o r o n o x i d e t h i c k n e s s , gas p r e s s u r e , s o l i d / l i q u i d i n t e r f a c e shape, v e r t i c a l t e m p e r a t u r e g r a d i e n t s and o t h e r s . The r e s u l t s show t h a t the t e m p e r a t u r e d i s t r i b u t i o n i n t h e gas s u r r o u n d i n g t h e c r y s t a l , and t h e b o r o n o x i d e t h i c k n e s s , were c r i t i c a l f a c t o r s i n d e t e r m i n i n g the d i s l o c a t i o n d e n s i t y and d i s t r i b u t i o n i n the c r y s t a l . The c r y s t a l r a d i u s , c r y s t a l l e n g t h and i n t e r f a c e c u r v a t u r e a l s o s t r o n g l y i n f l u e n c e d t h e d i s l o c a t i o n c o n f i g u r a t i o n . A f t e r c r y s t a l g rowth, t h e d i s l o c a t i o n d e n s i t y a t t h e end of the c r y s t a l was s t r o n g l y i n f l u e n c e d by t h e c o o l i n g p r o c e d u r e a d o p t e d . The d i s l o c a t i o n d i s t r i b u t i o n on c r o s s - s e c t i o n s o f t h e c r y s t a l e x h i b i t e d t w o - f o l d , f o u r - f o l d and e i g h t - f o l d symmetry d e p e n d i n g on growth and c o o l i n g c o n d i t i o n s and p o s i t i o n i n t h e c r y s t a l . I l l TABLE OF CONTENTS Page ABSTRACT II TABLE OF CONTENTS I l l LIST OF FIGURES XI LIST OF TABLES XXV LIST OF SYMBOLS XXVI ACKNOWLEDGEMENTS XXX CHAPTER 1 INTRODUCTION 1 CHAPTER 2 THE GROWTH OF BULK GaAs CRYSTALS 4 2.1 The L i q u i d E n c a p s u l a t e d C z o c h r a l s k i Technque ( LEC ) 4 2.2 C l a s s i f i c a t i o n of D e f e c t s i n LEC grown GaAs c r y s t a l 7 2.3 D i s l o c a t i o n s i n LEC-GaAs 8 2.4 E f f e c t o f D i s l o c a t i o n s on P r o p e r t i e s of GaAs D e v i c e s 11 CHAPTER 3 THE ORIGIN OF DISLOCATIONS IN LEC GaAs CRYSTALS 12 CHAPTER 4 DISLOCATION DENSITY AND CRYSTAL GROWTH 17 IV 4.1 S t r e s s e s i n t h e C r y s t a l Due to Thermal G r a d i e n t s u 17 4.1.1 C r y s t a l D i m e n s i o n s . 18 4.1.2 T h e r m a l C o n d i t i o n s D u r i n g C r y s t a l Growth 20 4.2 A l l o y A d d i t i o n to the GaAs 22 CHAPTER 5 MODELS OF DISLOCATION GENERATION IN GaAs 29 5 . 1 S t r e s s F i e l d s . 29 5.2 C a l c u l a t i o n of D i s l o c a t i o n D e n s i t i e s 36 CHAPTER 6 OBJECTIVES 40 CHAPTER 7 FORMULATION OF THE MODELS 43 7.1 Model of c r y s t a l growth 45 7.1.1 T e m p e r a t u r e F i e l d 45 7.1.1.1 G o v e r n i n g E q u a t i o n s 45 7.1.1.2 F i n i t e E l e m e n t E q u a t i o n s 48 7.1.2 S t r e s s F i e l d Model 53 7.1.2.1 G o v e r n i n g and Ele m e n t E q u a t i o n s 53 7.1.2.2 R e q u i r e m e n t s f o r t h e i n t e r p o l a t i o n f u n c t i o n s . . . . 57 7.1.2.3 F o r m u l a s f o r S t r e s s C a l c u l a t i o n s 59 7.1.2.4 L i n e a r E l e m e n t s 60 7.1.2.5 Q u a d r a t i c E l e m e n t 61 7.1.3 Von M i s e s and R e s o l v e d Shear S t r e s s e s 63 7.2 M o d e l l i n g f o r C o o l i n g A f t e r growth 67 V 7.3 A n a l y t i c a l S o l u t i o n s 71 7.3.1 A n a l y t i c a l QSS T e m p e r a t u r e F i e l d 71 7.3.2 A n a l y t i c a l s o l u t i o n s f o r the S t r e s s F i e l d 73 7.3.2.1 P l a n e S t r a i n A p p r o x i m a t i o n 73 7.3.2.2 A x i s y m m e t r i c S o l u t i o n s 74 CHAPTER 8 EVALUATION OF THE MODELS ' 76 8.1 Programming and I n p u t P a r a m e t e r s 76 8.1.1 I n p u t P a r a m e t e r s 86 8.1.2 N u m e r i c a l E v a l u a t i o n o f the A n a l y t i c a l S o l u t i o n s 93 8.2 C o m p a r i s o n o f Model P r e d i c t i o n s w i t h A n a l y t i c a l S o l u t i o n s 93 8.2.1 T e m p e r a t u r e F i e l d 93 8.2.2 S t r e s s F i e l d s 97 8.3 C o m p a r i s o n of Model P r e d i c t i o n s w i t h T e m p e r a t u r e Measurements 107 8.4 The T e m p e r a t u r e Model f o r C o o l i n g - Programming and C o n v e r g e n c y 115 CHAPTER 9 RESULTS AND ANALYSIS 120 9.1 Cone A n g l e 121 9.1.1 E f f e c t o f Cone A n g l e on S t r e s s Symmetry 133 9.1.2 E f f e c t of T h e r m a l C o n d i t i o n s 146 9.1.3 E f f e c t of t h e Heat T r a n s f e r C o e f f i c i e n t 155 9.1.4 E f f e c t o f N o n - l i n e a r i t y i n the T e m p e r a t u r e P r o f i l e 161 9.2 C r y s t a l L e n g t h 170 VI 9.2.1 E f f e c t o f c r y s t a l L e n g t h on S t r e s s Symmetry 175 9.3 C r y s t a l R a d i u s 190 9.3.1 E f f e c t o f R a d i u s on S t r e s s Symmetry 197 9.4 Growth V e l o c i t y 199 9.5 T h e r m a l C o n d i t i o n s 202 9.5.1 R a d i u s 27.5 mm 205 9.5.1.1 Boron o x i d e t h i c k n e s s 21.0 mm 205 9.5.1.2 B o r o n o x i d e t h i c k n e s s 40.0 mm 207 9.5.2 R a d i u s 40.0 mm 217 9.5.2.1 B o r o n o x i d e t h i c k n e s s 21 mm 217 9.5.2.2 Boron o x i d e t h i c k n e s s 40 mm 217 9.5.2.3 Boron O x i d e t h i c k n e s s 50 mm 221 9.6 Gas P r e s s u r e and C o m p o s i t i o n 227 9.7 C u r v a t u r e of the S o l i d - L i q u i d I n t e r f a c e 235 9.7.1 Convex i n t e r f a c e 236 9.7.1.1 C r y s t a l L e n g t h 236 9.7.1.2 E f f e c t of E n c a p s u l a n t T h i c k n e s s and G r a d i e n t 245 9.7.1.3 E f f e c t of c r y s t a l R a d i u s 251 9.7.2 Concave I n t e r f a c e 253 CHAPTER 10 RESULTS FOR COOLING AFTER GROWTH 257 10.1 Ambient T e m p e r a t u r e 1000°C 258 10.1.1. A r g o n 258 10.1.1.1 I n i t i a l G r a d i e n t GC = 70°C/cm 258 10.1.1.2 I n i t i a l G r a d i e n t GC/2 = 35°C/cm 269 10.1.1.3 I n i t i a l G r a d i e n t GC/4 = 17.5°C/cm 276 VII 10.1.2. Boron O x i d e 278 10.2 Ambient T e m p e r a t u r e 800°C/cm 283 10.3 A n a l y s i s o f t h e R e s u l t s f o r C o o l i n g 283 CHAPTER 11 SUMMARY AND CONCLUSIONS 290 REFERENCES 306 APPENDIX I EFFECT OF DISLOCATIONS ON GaAs AND DEVICES 320 1.1 E f f e c t o f D i s l o c a t i o n on P r o p e r t i e s 320 1.2 D i s l o c a t i o n s and F e r m i L e v e l 327 APPENDIX II I n t e g r a t i o n by P a r t s o f t h e Element E q u a t i o n f o r t h e T e m p e r a t u r e F i e l d 330 APPENDIX I I I E v a l u a t i o n o f t h e M a t r i x E l e m e n t s f o r t h e T e m p e r a t u r e F i e l d 332 APPENDIX IV E v a l u a t i o n o f S t i f f n e s s M a t r i x E l e m e n t s f o r L i n e a r E l e m e n t s " 338 APPENDIX V Q u a d r a t i c E l e m e n t C a l c u l a t i o n s 340 V I I I APPENDIX VI C a l c u l a t i o n o f R e s o l v e d S h e a r S t r e s s e s 353 APPENDIX VII T e m p e r a t u r e F i e l d D u r i n g C o o l i n g 362 APPENDIX V I I I A n a l y t i c a l S o l u t i o n o f t h e T e m p e r a t u r e F i e l d 367 APPENDIX IX P l a n e S t r a i n A n a l y t i c a l S t r e s s e s 372 APPENDIX X ANALYTICAL AS ISYMMETRIC SOLUTIONS FOR STRESSES 375 X . l A x i s y m m e t r i c T e m p e r a t u r e F i e l d 375 2 X. 2 A n a l y t i c a l A x i s y m m e t r i c S o l u t i o n f o r 6 = - p ... 385 APPENDIX XI COMPUTER PROGRAMS 390 XI . 1 Mesh G e n e r a t o r 390 XI.2 F i n i t e L i n e a r E l e m e n t Program f o r t h e T e m p e r a t u r e C a l c u l a t i o n s d u r i n g Growth 394 XI.3 F i n i t e L i n e a r E l e m e n t Program f o r the S t r e s s C a l c u l a t i o n s 402 XI.4 F i n i t e Q u a d r a t i c E l e m e n t Program f o r t h e S t r e s s C a l c u l a t i o n s 412 XI.5 Program f o r t h e C a l c u l a t i o n s of t h e MRSS-CRSS i n (010) S e c t i o n s 425 XI.6 Program f o r the C a l c u l a t i o n s o f MRSS-CRSS i n (001) S e c t i o n s 428 IX XI. 7 Program f o r the N u m e r i c a l E v a l u a t i o n of T e m p e r a t u r e s D u r i n g C o o l i n g 432 APPENDIX X I I I X I I . 1 Program f o r t h e N u m e r i c a l E v a l u a t i o n o f A n a l y t i c a l A x i s y m m e t r i c T e m p e r a t u r e F i e l d s 437 X I I . 2 Program f o r t h e N u m e r i c a l E v a l u a t i o n o f A n a l y t i c a l A n a l y t i c a l A x i s y m m e t r i c S t r e s s e s f o r R a d i a l T e m p e r a t u r e F i e l d s 439 X I I . 3 Program f o r t h e N u m e r i c a l E v a l u a t i o n o f A n a l y t i c a l A x i s y m m e t r i c S t r e s s e s f o r A x i s y m m e t r i c T e m p e r a t u r e F i e l d s 441 APPENDIX X I I I TEMPERATURE AND STRESS PLOTS FROM THE COMPUTER PROGRAMS OUTPUT (GROWTH) 444 XI11 . 1 Cone A n g l e CL, 10 mm ; R, 20 mm : B, 10 mm ; AP, 30 atm. ; BG, 100 C/cm ; AG, 50 C/cm 444 X I I I . 2 C r y s t a l L e n g t h R, 27.5 mm ; B.21 mm ; AP, 30 atm. ; CA, 30° ... 521 X I I I . 3 C r y s t a l R a d i u s R, 40 mm ; B, 21 mm ; CA, 30° ; AP, 30 atm 575 X I I I . 4 Growth V e l o c i t y R 27.5 mm ; CL , 55 mm ; CA, 30° ; B, 21 mm 617 X I I I . 5 T h e r m a l C o n d i t i o n R, 27.5 mm CL, 55 mm ; CA, 30 ; AP, 30 atm 624 X I I I . 6 T h e r m a l C o n d i t i o n s R, 40 mm CL, 80 mm ; CA, 30° ; AP , 30 atm 655 X I I I . 7 A r g o n P r e s s u r e , 2 a t m o s p h e r e s . R, 27.5 mm ; CL, 55 mm ; CA, 30° ; B, 21 mm .... 690 X I I I . 8 C u r v a t u r e o f the S o l i d / L i q u i d I n t e r f a c e CA. 30° ; AP, 30 atm 703 X APPENDIX XIV TEMPERATURE AND STRESS PLOTS FROM THE COMPUTER PROGRAMS OUTPUT (COOLING) 761 XIV.1 Ambient T e m p e r a t u r e 800°C, A r g o n 761 XIV.2 Ambient T e m p e r a t u r e 1000°C 799 XI L I S T OF FIGURES F i g . 2.1 S c h e m a t i c o f a LEC p u l l i n g chamber 5 F i g . 3.1 T r a n s m i s s i o n X - r a y t o p o g r a p h y i n GaAs - ( a ) and (b) undopgd ; ( c ) T e - d o p e d , ( 1 1 0 ) a x i a l s e c t i o n 13 F i g . 4.1 R a d i a l d i s l o c a t i o n d e n s i t y p r o f i l e s a c r o s s w a f e r s o b t a i n e d f r o m t h e f r o n t , m i d d l e , and t a i l of a c r y s t a l . The r a d i a l p r o f i l e s a r e "W" s h a p e d , and t h e a v e r a g e EPD i n c r e a s e s f r o m t h e f r o n t t o t h e t a i l 19 F i g . 4.2 Mean d e n s i t y o f " g r o w n - i n " d i s l o c a t i o n s i n GaAs s i n g l e c r y s t a l s (20-25 mm i n d i a m e t e r ) grown by t h e C z o c h r a l s k i LEC T e c h n i q u e , as a f u n c t i o n o f d o p a n t c o n c e n t r a t i o n : (1) Te, (2) Sn, (3) I n , (4) Zn 23 F i g . 4.3 H y p o t h e t i c a l d i s t r i b u t i o n o f t h e d o p a n t ( I n ) a l o n g t h e p u l l i n g a x i s o f t h e i n g o t . The i n d i u m c o n c e n t r a t i o n o s c i l l a t e s a r o u n d a mean v a l u e w h i c h i s p r o p o r t i o n a l t o t h e d i s t a n c e f r o m t h e t o p o f t h e i n g o t 26 F i g . 5.1 (a) and (b) - I s o t h e r m s and s h e a r s t r e s s t o p o g r a p h y i n two g a l l i u m a r s e n i d e s i n g l e c r y s t a l grown u n d e r d i f f e r e n t c o n d i t i o n s ( c ) - D i s t r i b u t i o n o f S h e a r S t r e s s e s T ̂  T , and (d) - d i s l o c a t i o n s d e n s i t y o v e r t h e c r o s s - s e c t i o n s o v e r g a l i u m a r s e n i d e s i n g l e c r y s t a l s . . . 30 F i g . 5.2 T R S S c o n t o u r s f o r t h e t o p w a f e r o f a <001> GaAs b o u l e 33 F i g . 5.3 D i s t r i b u t i o n o f t h e c a l c u l a t e d d i s l o c a t i o n d e n s i t y i n a g a l l i u m a r s e n i d e s i n g l e c r y s t a l b e i n g grown i n t h e < 111 > d i r e c t i o n f o r s l i p s y s t e m s w i t h B u r g e r s d i s l o c a t i o n v e c t o r s p e r p e n d i c u l a r 34 X I I F i g . 7.1 F l o w c h a r t o f t h e model u s e d t o d e r i v e t h e s t r e s s f i e l d s 44 F i g . 7.2 C r y s t a l c o n f i g u r a t i o n and c o o r d i n a t e s y s t e m u s e d i n t h e m a t h e m a t i c a l model 47 F i g . 8.1 F l o w c h a r t o f t h e c o m p u t e r p r o g r a m t o c a l c u l a t e t h e t e m p e r a t u r e f i e l d i n t h e c r y s t a l u s i n g a f i n i t e e l e m e n t method 78 F i g . 8.2 F l o w c h a r t o f t h e c o m p u t e r p r o g r a m f o r t h e s t r e s s c a l c u l a t i o n s u s i n g f i n i t e l i n e a r and q u a d r a t i c e l e m e n t s 81 F i g . 8.3 F l o w c h a r t o f t h e mesh g e n e r a t o r c o m p u t e r p r o g r a m 84 F i g . 8.4 F l o w c h a r t o f t h e c o m p u t e r p r o g r a m t o c a l c u l a t e and p l o t RSS i n a v e r t i c a l (010) p l a n e 87 F i g . 8.5 F l o w c h a r t o f t h e c o m p u t e r p r o g r a m t o c a l c u l a t e and p l o t RSS i n a (001) p l a n e 88 F i g . 8.6 E s t i m a t e d r a d i a t i v e and c o n v e c t i o n h e a t t r a n s f e r c o e f f i c i e n t s f o r G a A s / B 2 0 ( 1 ) , He ( g ) , N 2 (g) and A (g) as a f u n c t i o n o f a m b i e n t t e m p e r a t u r e . The n u m e r i c a l l a b e l s a r e t h e p r o d u c t o f t h e c a r r i e r c o n c e n t r a t i o n X c r y s t a l d i a m e t e r i n u n i t s o f cm 90 F i g . 8.7 T o t a l h e a t t r a n s f e r c o e f f i c i e n t h, i n B2®3 an(* a r g o n as a f u n c t i o n o f t h e a m b i e n t t e m p e r a t u r e T (1) T o t a l h e a t t r a n s f e r c o e f f i c i e n t i n ^2°3' ( § ) T o t a l h e a t t r a n s f e r c o e f f i c i e n t i n a r g o n p r e s s u r i z e d a t 30 atm 90 F i g . 8.8 T e m p e r a t u r e d e p e n d e n c e o f t h e c r i t i c a l s t r e s s f o r d i s l o c a t i o n g e n e r a t i o n i n GaAs : (1) Te-doped m a t e r i a l , n = 2 X 10*® cm (2) Te-doped m a t e r i a l , n = 7 X 10 cm X I I I (3) Zn-doped m a t e r i a l , (4) undoped m a t e r i a l . . 9 X 1 0 1 8 cm 3 92 F i g . 8.9 A c o m p a r i s o n of f i n i t e e l e m e n t and a n a l y t i c a l c a l c u l a t e d t e m p e r a t u r e c u r v e s f o r h 0.3 cm 94 F i g . 8.10 A c o m p a r i s o n o f f i n i t e e l e m e n t and a n a l y t i c a l c a l c u l a t e d t e m p e r a t u r e c u r v e s f o r h 0.6 cm" 95 F i g . 8.11 Mesh e m p l o y e d i n t h e c a l c u l a t i o n s o f t h e t e m p e r a t u r e f i e l d s shown i n F i g . 8.9 and 8.10 Number of nodes = 45. Number o f e l e m e n t s = 64 96 F i g . 8.12 C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r / r f o r r a d i a l t e m p e r a t u r e f i e l d s (1) F i n i t e e l e m e n t w i t h a v e r a g e d e l e m e n t (2) F i n i t e e l e m e n t w i t h n o d a l t e m p e r a t u r e s (3) A n a l y t i c a l p l a n e s t r a i n (4) Ana 1 y t i c a 1 - a x i s y m m e t r i c 98 F i g . 8.13 C a l c u l a t e d a z i m u t h a l s t r e s s e s as a f u n c t i o n o f r / r f o r r a d i a l t e m p e r a t u r e f i e l d s (1) F i n i t e e l e m e n t w i t h a v e r a g e d e l e m e n t t e m p e r a t u r e s (2) F i n i t e e l e m e n t w i t h n o d a l t e m p e r a t u r e s (3) Ana 1 y t i c a 1 - p 1 a n e s t r a i n (4) A n a l y t i c a l - a x i s y m m e t r i c 99 F i g . 8.14 C a l c u l a t e d a x i a l s t r e s s e s as a f u n c t i o n o f r / r f o r r a d i a l t e m p e r a t u r e f i e l d s (1) F i n i t e e l e m e n t w i t h a v e r a g e d e l e m e n t t e m p e r a t u r e s (2) F i n i t e e l e m e n t w i t h n o d a l t e m p e r a t u r e s (3) Ana 1 y t i c a l - p i a n e s t r a i n (4) A n a l y t i c a l - a x i s y m m e t r i c 100 F i g . 8.15 F o u r s t e p s i n t h e mesh r e f i n e m e n t u s e d t o a n a l y s e t h e c o n v e r g e n c y o f t h e f i n i t e e l e m e n t s t r e s s c a l c u l a t i o n s . (a) NN = 15, NE = 16 (b) NN = 45, NE = 16 ( c ) NN = 45, NE = 64. (d) NN = 153, NE = 256 (b) Q u a d r a t i c e l e m e n t s ( a ) , ( c ) and (d) l i n e a r e l e m e n t s 102 F i g . 8.16 C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r / r f o r d i f f e r e n t numbers of nodes and s i z e s o f e l e m e n t (1) F i n i t e e l e m e n t w i t h a v e r a g e d XIV e l e m e n t t e m p e r a t u r e (2) F i n i t e e l e m e n t w i t h n o d a l t e m p e r a t u r e s . L = L i n e a r e l e m e n t , Q = Q u a d r a t i c e l e m e n t 103 F i g . 8.17 C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r / r f o r a x i s y m m e t r i c t h e r m a l f i e l d s (l)°Finite e l e m e n t w i t h a v e r a g e d e l e m e n t t e m p e r a t u r e s (2) F i n i t e e l e m e n t w i t h n o d a l t e m p e r a t u r e s (3) Ana 1 y t i c a 1 - p 1 a n e s t r a i n (4) A n a l y t i c a l - a x i s y m m e t r i c 105 F i g . 8.18 P o s i t i o n o f t h e r m o c o u p l e s i n GaAs c r y s t a l . B = B o r i c o x i d e l a y e r , a r g o n p r e s s u r e 3.04 MPa. R e f e r e n c e 218 108 F i g . 8.19 T e m p e r a t u r e s measured w i t h t h e r m o c o u p l e s 2, 3 and 4 i n F i g u r e 8.18 as a f u n c t i o n o f the r e l a t i v e p o s i t i o n of the t h e r m o c o u p l e s w i t h t h e i n t e r f a c e . R e f e r e n c e 218 110 F i g . 8.20 M e a s u r e d and c a l c u l a t e d t e m p e r a t u r e s a l o n g t h e c r y s t a l a x i s a t f o u r c r y s t a l l e n g t h s I l l F i g . 8.21 M e a s u r e d and c a l c u l a t e d t e m p e r a t u r e s a d j a c e n t to th e o u t s i d e s u r f a c e o f t h e c r y s t a l a t f o u r c r y s t a l l e n g t h s . The measured ambient t e m p e r a t u r e i s a l s o shown 112 F i g . 8.22 M e a s u r e d and c a l c u l a t e d a x i a l t e m p e r a t u r e g r a d i e n t s a l o n g the c r y s t a l a x i s a t f o u r c r y s t a l 1 e n g t h s 113 F i g . 8.23 Flow c h a r t of t h e computer program f o r the n u m e r i c a l e v a l u a t i o n o f the a n a l y t i c a l t e m p e r a t u r e f i e l d s d u r i n g c o o l i n g o f the c r y s t a l 117 F i g . 8.24 (a) T y p i c a l t e m p e r a t u g e Q f i e 1 d o b t a i n e d d u r i n g c o o l i n g u n i t s a r e 10 C. (b) and ( c ) Von M i s e s s t r e s s c o n t o u r s (MPa) f o r the t e m p e r a t u r e f i e l d g i v e n i n ( a ) . (b) NN = 451, NE = 800. (c ) NN = 1105, NE = 2048 118 Von M i s e s S t r e s s c o n t o u r s (MPa) i n v e r t i c a l p l a n e s f o r f i v e cone a n g l e s , (a) 7 . 1 ° , (b) 3 0 ° , ( c ) 4 5 ° , (d) 54. 7 ° , (e) 6 5 ° . Cone s u r f a c e i n (d) c o i n c i d e s w i t h a (111) p l a n e . C r y s t a l r a d i u s , 20 mm ; c r y s t a l l e n g t h , 10 mm ; B 2 ^ 3 t h i c k n e s s , 10 ram ; B 0^ g r a d i e n t , 100 C/cm ; a r g o n p r e s s u r e , 30 atm. ; a r g o n g r a d i e n t , 50°C/cm Maximum r e s o l v e d s h e a r s t r e s s (MRSS) c o n t o u r s i n MPa, i n v e r t i c a l (010) p l a n e s f o r f i v e cone a n g l e s . (a) 7.1°, (b) 3 0 ° , ( c ) 4 5 ° , (d) 5 4 . 7 ° , (e) 65 . Cone s u r f a c e i n (d) c o i n c i d e s w i t h a (111) p l a n e . C o n d i t i o n s a r e the same as i n F i g u r e 9.1 T o t a l r e s o l v e d s h e a r s t r e s s (TRSS) c o n t o u r s i n MPa f o r t h e 45 cone a n g l e c r y s t a l . Compare t h e l a r g e s t r e s s l e v e l s of the TRSS w i t h the s t r e s s l e v e l s f o r same c r y s t a l shown i n F i g u r e 9 . 1 ( c ) f o r the VMS and F i g u r e 9.2(c) f o r t h e MRSS C o n t o u r s o f t h e MRSS (MPa) i n e x c e s s of (a) CRSS ( y i e l d ) ; (b) CRSS (MB) ; ( c ) CRSS (MBTe). Shaded r e g i o n s i n d i c a t e a r e a s i n w h i c h t h e MRSS i s l e s s t h a n t h e CRSS S l i p mode d i s t r i b u t i o n i n the (010) p l a n e f o r the 45 cone a n g l e c r y s t a l c o r r e s p o n d i n g t o t h e MRSS d i s t r i b u t i o n shown i n F i g u r e 9 . 2 ( c ) C o n t o u r s i n MPa o f the MRSS i n e x c e s s of (a) CRSS ( Y i e l d ) ; (b) CRSS (MB) and ( c ) CRSS (MBTe). In most of t h e c r y s t a l the MRSS i s l a r g e r t h a n t h e c r i t i c a l v a l u e s . The bump shaped c o n t o u r s (A) g i v e complex s t r e s s d i s t r i b u t i o n s i n p e r p e n d i c u l a r c r o s s - s e c t i o n s MRSS c o n t o u r s (MPa) i n c r o s s - s e c t i o n s p e r p e n - d i c u l a r t o t h e c r y s t a l a x i s f o r f i v e cone a n g l e s , (a) 7 . 1 ° , (b) 3 0 ° , ( c ) 4 5 ° , (d) 5 4 . 7 ° , (e) 6 5 ° . S e c t i o n (a-d) a r e 7.5 mm from the i n t e r f a c e and s e c t i o n (e) i s 8.0 mm from the i n t e r f a c e . The h o r i z o n t a l d i r e c t i o n c o r r e s p o n d s to the [100] d i r e c t i o n and t h e v e r t i c a l d i r e c t i o n c o r r e s - ponds to t h e [010] d i r e c t i o n XVI F i g . 9.8 C o n t o u r s of the MRSS-CRSS ( Y i e l d ) (MPa) f o r th e 65 cone a n g l e c r y s t a l . MRSS c o n t o u r s a r e shown i n F i g u r e 9 . 7 ( e ) . At h i g h cone a n g l e s MRSS l e v e l s a r e l a r g e r t h a n CRSS a t the c e n t r e and o u t s i d e p a r t o f t h e wafer 142 F i g . 9.9 S t r e s s c o n t o u r s (MPa) i n c r o s s - s e c t i o n 2.5 mm from the i n t e r f a c e , (a) and (b) i n a 7.1° cone a n g l e c r y s t a l , (c) i n a 45° cone a n g l e c r y s t a l , (a) MRSS c o n t o u r s , (b) and ( c ) MRSS-CRSS ( Y i e l d ) . F o r t h e 7.1 cone a n g l e t h e r e i s e i g h t - f o l d symmetry at the c e n t r e i n (a) which i s not seen i n (b) b e c a u s e s t r e s s l e v e l s a r e l e s s t h a n CRSS. F o r the 45 cone a n g l e t h e r e i s f o u r - f o l d symmetry 145 F i g . 9.10 S l i p mode d i s t r i b u t i o n i n a (001) p l a n e c o r r e s - p o n d i n g to the MRSS d i s t r i b u t i o n shown i n F i g u r e 9 . 7 ( c ) f o r t h e 45 cone a n g l e c r y s t a l a t 7.5 mm from t h e i n t e r f a c e . The e i g h t - f o l d d i s t r i b u t i o n o f t h e mode a t t h e edge of t h e s e c t i o n i s a s s o c i a t e d w i t h the e i g h t - f o l d symmetry o f t h e MRSS 147 F i g . 9.11 T e m p e r a t u r e d i s t r i b u t i o n as a f u n c t i o n of e x t e r n a l t e m p e r a t u r e g r a d i e n t f o r a 45° cone a n g l e c r y s t a l . T h e r m al g r a d i e n t s i n the b o r o n o x i d e a r e : (a) 50 C/cm, (b) 100 C/cm, ( c ) 200 C/cm a n g Q ( d ) 400 C/cm. T e m p e r a t u r e s a r e g i v e n i n 10 C. F o r low g r a d i e n t s i s o t h e r m s a r e n e a r l y f l a t and s l i g h t l y c o nvex. F o r l a r g e r g r a d i e n t s i s o t h e r m s a r e c u r v e d and c o n c a v e 149 F i g . 9.12 S t r e s s c o n t o u r s (MPa) i n a (010) p l a n e f o r a 45° cone a n g l e c r y s t a l grown w i t h a 50°C/cm g r a d i e n t i n the b o r o n o x i d e , (a) MRSS c o n t o u r s does not show t h e bump shaped s t r e s s d i s t r i b u t i o n below the s h o u l d e r i n ( a ) . The MRSS-CRSS ( Y i e l d ) i s p o s i t i v e o n l y i n a few r e g i o n s i n (b) 151 F i g . 9.13 (a) MRSS c o n t o u r s and (b) MRSS-CRSS ( Y i e l d ) c o n t o u r s f o r a 200 C/cm g r a d i e n t i n the b o r o n o x i d e . U n i t s a r e i n MPa. MRSS s t r e s s a r e l a r g e below t h e s h o u l d e r i n ( a ) . The MRSS-CRSS i s g r e a t e r t h a n z e r o i n most of the c r y s t a l i n ( b ) . 152 F i g . 9.14 (a) MRSS c o n t o u r s , (b) MRSS-CRSS ( Y i e l d ) c o n t o u r s , ( c ) S l i p mode d i s t r i b u t i o n . G r a d i e n t XVII i n the b o r o n o x i d e i s 400°C/cm. In (a) c o n t o u r s a r e s i m i l a r to t h o s e o b t a i n e d f o r a 200°C/cm g r a d i e n t . S t r e s s l e v e l s have d o u b l e d . In (b) o n l y a s m a l l a r e a i n the seed d e v e l o p e d s t r e s s e s l e s s t h a n the CRSS. In ( c ) the mode d i s t r i b u t i o n i s s i m i l a r t o t h a t shown i n F i g u r e 9.5 f o r a 100°C/cm g r a d i e n t 154 F i g . 9.15 MRSS c o n t o u r s i n MPa i n a c r o s s - s e c t i o n a t 7.5 mm from t h e i n t e r f a c e f o r t h e c r y s t a l shown i n F i g u r e 9 . 1 4 ( a ) . The symmetry i s s i m i l a r to t h a t shown i n F i g u r e 9.7(c) f o r a 100°C/cm g r a d i e n t 156 F i g . 9.16 T e m p e r a t u r e p r o f i l e s at the s u r f a c e o f 45° cone a n g l e c r y s t a l s f o r t h r e e d i f e r e n t c o n d i t i o n s . C urve H c o r r e s p o n d s to t h e t e m p e r a t u r e s c a l c u - l a t e d u s i n g t h e o r i g i n a l h e a t t r a n s f e r c o e f - f i c i e n t v a l u e s . In c u r v e s H / 1.3 and H x 1.3 the o r i g i n a l h e a t t r a n s f e r c o e f f i c i e n t v a l u e s were d i v i d e d and m u l t i p l i e d by 1.3 r e s p e c t i v e l y 158 F i g . 9.17 S t r e s s c o n t o u r s i n MPa d e r i v e d from t h e t e m p e r a t u r e f i e l d o b t a i n e d u s i n g h e a t t r a n s f e r c o e f f i c i e n t v a l u e s 1.3 l a r g e r t h a n o r i g i n a l v a l u e s . (a) MRSS c o n t o u r s . (b) MRSS-CRSS ( Y i e l d ) 159 F i g . 9.18 S t r e s s c o n t o u r s i n MPa d e r i v e d from, the t e m p e r a t u r e f i e l d o b t a i n e d u s i n g h e a t t r a n s f e r c o e f f i c i e n t v a l u e s 1.3 t i m e s s m a l l e r t h a n o r i g i n a l v a l u e s . (a) MRSS c o n t o u r s . (b) MRSS-CRSS ( Y i e l d ) . V a r i a t i o n s of o n l y 20 * a r e o b s e r v e d i n t h e maximum MRSS v a l u e s between (a) and F i g u r e 9.17(a) 160 F i g . 9.19 MRSS (MPa) c o n t o u r s f o r a c r y s t a l g r o w i n g under f o u r d i f f e r e n t c o n d i t i o n s g i v e n i n T a b l e 9.3. (a) Run #1 ; (b) Run #2 ; ( c ) Run #3 ; (d) Run #4. From (a) t o (b) the AMRSS d o u b l e s . In (d) th e bump shape below the s h o u l d e r does not app e a r 165 F i g . 9.20 MRSS (MPa) c o n t o u r s i n a (001) p l a n e at a d i s t a n c e o f 5.0 mm from the i n t e r f a c e i n th e c r y s t a l shown i n F i g u r e 9.19(d) ( r u n #4) XVI I I The f o u r - f o l d symmetry w i t h a s l i g h t t w o - f o l d symmetry i s o b s e r v e d 167 F i g . 9.21 MRSS-CRSS ( Y i e l d ) c o n t o u r s (MPa) f o r t h e f o u r g r o w t h c o n d i t i o n s shown i n T a b l e 9.3 and MRSS c o n t o u r s shown i n F i g u r e 9.19 (a) Run #1 ; (b) Run #2 ; ( c ) Run #3 ; (d) Run #4 169 F i g . 9.22 MRSS (MPa) c o n t o u r s f o r f i v e c r y s t a l l e n g t h s . (a) 13.75 mm ; (b) 27.5 mm ; (c) 55.0 mm ; (d) 82.5 mm ; (e) 110.0 mm . C r y s t a l r a d i u s i s 27.5 mm 172 F i g . 9.23 MRSS-CRSS ( y i e l d ) (MPa) f o r f i v e c r y s t a l l e n g t h s . (a) 13.75 mm ; (b) 27.5 mm ; ( c ) 55.0 mm ; (d) 82.5 mm ; (e) 110.0 mm . C r y s t a l r a d i u s i s 27.5 mm 174 F i g . 9.24 S l i p mode of the MRSS f o r the f i v e c r y s t a l l e n g t h s shown i n F i g u r e 9.22 (a) 13.75 mm ; (b) 27.5 mm ; ( c ) 55.0 mm ; (d) 82.5 mm ; (e) 110.0 mm . C r y s t a l r a d i u s i s 27.5 mm 176 F i g . 9.25 MRSS-CRSS ( Y i e l d ) (MPa) i n a (001) p l a n e a t a d i s t a n c e of 8.25 mm from t h e cone a t f o u r c r y s t a l l e n g t h s , (a) 13.75 mm ; (b) 27.5 mm ; ( c ) 55.0 mm ; (d) 82.5 mm. C r y s t a l r a d i u s 27.5 mm 182 F i g . 9.26 MRSS-CRSS ( Y i e l d ) (MPa) i n a (001) p l a n e a t a d i s t a n c e o f 49.5 mm from the cone at two c r y s t a l l e n g t h s , (a) 55.0 mm ; (b) 82.5 mm. C r y s t a l r a d i u s 27.5 mm 185 F i g . 9.27 MRSS-CRSS ( Y i e l d ) (MPa) i n a (001) p l a n e a t a d i s t a n c e o f 77.0 mm from the cone a t two c r y s t a l l e n g t h s , (a) 82.5 mm ; (b) 110.0 mm. C r y s t a l r a d i u s 27.5 mm 188 F i g . 9.28 MRSS (MPa) c o n t o u r s f o r f o u r c r y s t a l l e n g t h s . (a) 20.0 mm ; (b) 40.0 mm ; ( c ) 80.0 mm ; (d) 100.00 mm. C r y s t a l R a d i u s 40.0 mm 193 F i g . 9.29 (a) A x i a l a l o n g the c r y s t a l a x i s and (b) r a d i a l t h e r m a l g r a d i e n t s as a f u n c t i o n o f d i s t a n c e XIX from the i n t e r f a c e f o r (1) f o r a r a d i u s of 27 (3) f o r a r a d i u s of 40 (1) 55.0 mm ; (2 ) 40.0 two c r y s t a l r a d i u s . Curve 5 mm ; and c u r v e s (2) and 0 mm. C r y s t a l l e n g t h s a r e mm and (3) 80.0 mm 194 F i g . 9.30 MRSS-CRSS ( Y i e l d ) (MPa) f o r f o u r c r y s t a l l e n g t h s . (a) 20.0 mm ; (b) 40.0 mm ; ( c ) 80.0 mm ; (d) 100.00 mm. C r y s t a l R a d i u s 40.0 mm 196 F i g . 9.31 T e m p e r a t u r e f i e l d s f o r t h r e e growth v e l o c i t i e s (a) 0.0001 cm/s, (b) 0.001 cm/s and ( c ) 0.01 cm/s. T e m p e r a t u r e g i v e n i n 10 °C. L i t t l e change i s o b s e r v e d from (a) to ( b ) . From (b) to ( c ) g r a d i e n t s have i n c r e a s e d . R a d i u s 27.5 mm, l e n g t h 55 mm, e n c a p s u l a n t t h i c k n e s s 21 mm 201 F i g . 9.32 MRSS-CRSS ( Y i e l d ) (MPa) f o r t h e t h r e e t e m p e r a - t u r e f i e l d s shown i n F i g u r e 9 . 3 1 ( a - c ) c o r r e s - p o n d i n g t o t h r e e growth v e l o c i t i e s , (a) 0.0001 cm/s, (b) 0.001 cm/s and ( c ) 0.01 cm/s. In (a) and (b) t h e s t r e s s f i e l d s a r e s i m i l a r . In ( c ) the s t r e s s d i s t r i b u t i o n and s t r e s s v a l u e s change 204 F i g . 9.33 T e m p e r a t u r e p r o f i l e s a l o n g the c r y s t a l s u r f a c e i n t h e e n v i r o n m e n t s u r r o u n d i n g t h e c r y s t a l employed i n the c a l c u l a t i o n s . The p r o f i l e s a r e d e r i v e d from F i g u r e 8.19, c u r v e 4, r e p r o d u c e d as c u r v e G. The v a l u e s of g r a d i e n t a r e a v e r a g e d i n t h e c r y s t a l l e n g t h c o n s i d e r e d . E n c a p s u l a n t t h i c k n e s s 21 mm 206 F i g . 9.34 MRSS-CRSS ( Y i e l d ) (MPa) f o r f o u r t e m p e r a t u r e p r o f i l e s w i t h a v e r a g e g r a d i e n t s (a) 33°c/cm, (b) 17°C/cm, ( c ) ll°C/cm and (d) 8°C/cm. R a d i u s 27.5 mm, l e n g t h 55 mm. E n c a p s u l a n t t h i c k n e s s 21 mm 209 F i g . 9.35 T e m p e r a t u r e p r o f i l e s used i n the c a l c u l a t i o n s f o r two b o r o n o x i d e t h i c k n e s s e s . (a) 40 mm, (b) 50 mm. The p r o f i l e s a r e d e r i v e d from c u r v e G. The v a l u e s of g r a d i e n t s g i v e n a r e a v e r a g e d a l o n g t h e c r y s t a l l e n g t h 211 F i g . 9.36 MRSS-CRSS ( Y i e l d ) (MPa) f o r t h r e e a v e r a g e g r a d i e n t s . (a) 58 C/cm, (b) 29°C/cm, ( c ) XX 19 C/cm. R a d i u s 27.5 mm, l e n g t h 55 mm, b o r o n o x i d e t h i c k n e s s 40 mm 213 F i g . 9.37 MRSS-CRSS ( Y i e l d ) (MPa) f o r t h r e e a v e r a g e g r a d i e n t s , (a) 50°c/cm, (b) 25°C/cm, ( c ) 17"C/ cm. R a d i u s 27.5 mm, l e n g t h 55 mm, b o r o n o x i d e t h i c k n e s s 50 mm 215 F i g . 9.38 MRSS-CRSS ( Y i e l d ) (MPa) f o r two a v e r a g e g r a d i - e n t s , (a) 14°C/cm, (b) 7 C/cm. R a d i u s 40 mm, l e n g t h 80 mm, b o r o n o x i d e t h i c k n e s s 21 mm 218 F i g . 9.39 MRSS-CRSS ( Y i e l d ) (MPa) f o r f o u r a v e r a g e g r a d i e n t s , (a) 46 C/cm, (b) 26°C/cm, ( c ) 15°C/cm, (d) 9°C/cm. R a d i u s 40 mm, l e n g t h 80 mm, boro n o x i d e t h i c k n e s s 40 mm 220 F i g . 9.40 MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l l e n g t h o f 52 mm c o m p a r a b l e t o t h e b o r o n o x i d e t h i c k n e s s o f 50 mm. R a d i u s 40 mm. A v e r a g e g r a d i e n t 50 C/cm... 222 F i g . 9.41 MRSS-CRSS ( Y i e l d ) (MPa) f o r t h r e e a v e r a g e g r a d i e n t s , (a) 43°C/cm, (b) 14°C/cm, ( c ) 7°C/cm. R a d i u s 40 mm, l e n g t h 80 mm, bor o n o x i d e t h i c k n e s s 50 mm 224 F i g . 9.42 MRSS-CRSS ( Y i e l d ) (MPa) f o r f o u r t e m p e r a t u r e p r o f i l e s w i t h a v e r a g e g r a d i e n t s , (a) 33 C/cm, (b) 17°C/cm, (c) 11 C/cm and (d) 8°C/cm. R a d i u s 27.5 mm, l e n g t h 55 mm. E n c a p s u l a n t t h i c k n e s s 21 mm. A r g o n p r e s s u r e 2 atm 229 F i g . 9.43 C o r r e l a t i o n between t h e measured t e m p e r a t u r e g r a d i e n t s a c r o s s t h e e n c a p s u l a n t and t h e h e a t t r a n s f e r c o e f f i c i e n t v a l u e s as a f u n c t i o n o f gas p r e s s u r e and gas n a t u r e . Curve A from R e f s . 143- 144 u s i n g c o n v e c t i o n from a v e r t i c a l w a l l ; and c u r v e B u s i n g c o n v e c t i o n from an h o r i z o n t a l s u r f a c e 234 F i g . 9.44 Element and n o d a l c o n f i g u r a t i o n a t t h e i n t e r f a c e f o r c r y s t a l s w i t h c u r v e d i n t e r f a c e , (a) Convex i n t e r f a c e , (b) Concave i n t e r f a c e 237 F i g . 9.45 F i g . 9.46 F i g . 9.47 F i g . 9.48 F i g . 9.49 F i g . 9.50 F i g . 9.51 F i g . 9.52 F i g . 9.53 XXI T e m p e r a t u r e (10 C) and Von N i s e s s t r e s s (MPa) f i e l d s f o r a c r y s t a l w i t h a convex i n t e r f a c e shape. R a d i u s 27.5 mm. L e n g t h 27.5 mm. E n c a p s u l a n t t h i c k n e s s 21 mm. T e m p e r a t u r e p r o f i l e as shown i n F i g u r e 8.19, c u r v e 4 239 MRSS (MPa) c o n t o u r s f o r a c r y s t a l w i t h convex i n t e r f a c e a t t h r e e l e n g t h s , (a) 27.5 mm, (b) 55 mm, ( c ) 82.5 mm. The l a r g e s t s t r e s s e s a t the i n t e r f a c e edge a r e (a) 7.22 MPa, (b) 9.19 MPa, ( c ) 9.66 MPa. R a d i u s 27.5 mm, Boron o x i d e t h i c k n e s s 21mm. P r o f i l e as in. F i g u r e 8.19, c u r v e 4 240 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s f o r a c r y s t a l w i t h convex i n t e r f a c e a t t h r e e l e n g t h s (a) 27.5 mm, (b) 55 mm, ( c ) 82.5 mm. R a d i u s 27.5 mm, bo r o n o x i d e t h i c k n e s s 21 mm. P r o f i l e as i n F i g u r e 8.19, c u r v e 4 242 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n a (001) p l a n e at a d i s t a n c e of 11 mm from t h e cone i n the c r y s t a l shown i n F i g u r e 9.47(a) 244 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n (001) p l a n e s at t h r e e d i s t a n c e s form the cone. (a) 79.75 mm, (b) 66.0 mm, (c) 52.25 mm. S e c t i o n c o r r e s p o n d s to the c r y s t a l shown i n F i g u r e 9.47(c) 248 (a) MRSS (MP) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l w i t h convex i n t e r f a c e . B o ron o x i d e t h i c k n e s s 50 mm. R a d i u s 27.5 mm. L e n g t h 55 mm. Av e r a g e g r a d i e n t 50°c/cm 249 (a) MRSS (MP) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l w i t h convex i n t e r f a c e . B o r o n o x i d e t h i c k n e s s 50 mm. R a d i u s 27.5 mm. L e n g t h 55 mm. Av e r a g e g r a d i e n t 17°C/cm 250 (a) MRSS (MP) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l w i t h convex i n t e r f a c e . Boron o x i d e t h i c k n e s s 21 mm. R a d i u s 40 mm. L e n g t h 80 mm. Av e r a g e g r a d i e n t 55°C/cm 252 (a) MRSS (MP) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l w i t h convex i n t e r f a c e . (a) R a d i u s XXI I 27.5 mm, (b) 40 mm. Boron o x i d e t h i c k n e s s 21 mm 254 F i g . 9.54 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n a (001) p l a n e a t a d i s t a n c e of 2.75 mm from t h e i n t e r f a c e i n the c r y s t a l shown i n F i g u r e 9.53(a) 256 F i g . 10.1 T e m p e r a t u r e ( 1 0 3 ° C ) f i e l d d u r i n g c o o l i n g a t f o u r t i m e s . (a) 5 s, (b) 10 s, ( c ) 20 s, (d) 60 s, I n i t i a l g r a d i e n t c l o s e t o i n t e r f a c e 70 C/cm. Argo n t e m p e r a t u r e 1000°C 260 F i g . 10.2 MRSS-CRSS ( Y i e l d ) (MPa) d u r i n g c o o l i n g f o r the f o u r t e m p e r a t u r e f i e l d s shown i n F i g u r e 1 0 . 1 ( a - d ) . (a) 5 s, (b£ 10 s, ( c ) 20 s, (d) 60 s. I n i t i a l g r a d i e n t 70 C/cm. Arg o n t e m p e r a t u r e 1000 C 263 F i g . 10.3 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n (001) p l a n e s at f o u r d i s t a n c e s from t h e bottom ( i n t e r f a c e ) i n the c r y s t a l shown i n F i g u r e 10.2(b) 10 s, (a) 2.75 mm, (b) 8.25 mm, ( c ) 16.5 mm, (d) 27.5 mm 268 F i g . 10.4 MRSS-CRSS ( Y i e l d ) (MPa) i n (001) p l a n e s a t two d i s t a n c e s from the bottom i n t h e c r y s t a l shown i n F i g u r e 10.2 (d) 60 s, (a) 2.75 mm ( b) 8.25 mm 271 F i g . 10.5 (a) T e m p e r a t u r e f i e l d ( 1 0 3 ° c ) and (b) MRSS-CRSS ( Y i e l d ) (MPa^ f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n at 1000 C a f t e r 10 s. I n i t i a l g r a d i e n t 35 C/cm 272 F i g . 10.6 (a) T e m p e r a t u r e f i e l d ( 1 0 3 o C ) and (b) MRSS-CRSS ( Y i e l d ) (MPa^ f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 1000 C a f t e r 60 s. I n i t i a l g r a d i e n t 35 C/cm 273 F i g . 10.7 MRSS-CRSS ( Y i e l d ) (MPa) i n (001) p l a n e s a t two d i s t a n c e s from t h e bottom i n t h e c r y s t a l shown i n F i g u r e 10.5 (b) 10 s. (a) 8.25 mm (b) 27.5 mm 275 F i g . 10.8 (a) T e m p e r a t u r e f i e l d ( 1 0 3 o C ) and (b) MRSS-CRSS XXI11 ( Y i e l d ) (MPa^ f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n at 1000 C a f t e r 10 s. I n i t i a l g r a d i e n t 17.5°C/cm 277 F i g . 10.9 MRSS-CRSS ( Y i e l d ) (MPa) i n (001) p l a n e s a d i s t a n c e of 27.5 mm from t h e bottom i n t h e c r y s t a l shown i n F i g u r e 10.8 (d) 10 s 279 F i g . 10.10 (a) T e m p e r a t u r e f i e l d ( 1 0 3 o C ) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n b o r o n o x i d e at 1000 C a f t e r 10 s. I n i t i a l g r a d i e n t OoC/cm 280 F i g . 10.11 (a) T e m p e r a t u r e f i e l d ( 1 0 3 ° C ) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n b o r o n o x i g e at 1000°C a f t e r 10 s. I n i t i a l g r a d i e n t 5oC/cm 281 F i g . 10.12 (a) T e m p e r a t u r e f i e l d (10 C) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n b o r o n o x i d e a t 1000°C a f t e r 10 s. I n i t i a l g r a d i e n t 17.5°C/cm 282 F i g . 10.13 (a) T e m p e r a t u r e f i e l d ( 1 0 3 o C ) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 800 C a f t e r 10 s. I n i t i a l g r a d i e n t 70 C/cm 284 F i g . 10.14 (a) T e m p e r a t u r e f i e l d ( 1 0 3 o C ) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 800 C a f t e r 10 s. I n i t i a l g r a d i e n t 35 C/cm . . 285 F i g . 10.15 (a) T e m p e r a t u r e f i e l d ( 1 0 3 o C ) and (b) MRSS-CRSS ( Y i e l d ) (^Pa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n at 8 OoC a f t e r 10 s. I n i t i a l g r a d i e n t 17.5°C/cm 286 F i g . 10.16 T e m p e r a t u r e g r o f i l e s i n the c r y s t a l c o o l i n g i n a r g o n a t 800 C f o r two i n i t i a l g r a d i e n t s i n t h e c r y s t a l . (a) 70°C/cm (b) 35°C/cm. The r i g h t p a r t c o r r e s p o n d s to t h e r a d i a l p r o f i l e s and the l e f t p a r t t o the a x i a l p r o f i l e . F u l l and b r o k e n l i n e s c o r r e s p o n d t o t h e a x i s and s u r f a c e t e m p e r a t u r e s r e s p e c t i v e l y 288 XXIV F i g . 10.17 T e m p e r a t u r e p r o f i l e s i n t h e c r y s t a l c o o l i n g i n b o r o n o x i d e a t 1000 C f o r two i n i t i a l g r a d i e n t s i n t h e c r y s t a l . (a) 70°C/cm (b) 35°C/cm. The r i g h t p a r t c o r r e s p o n d s to t h e r a d i a l p r o f i l e s and t h e l e f t p a r t t o t h e a x i a l p r o f i l e . F u l l and b r o k e n l i n e s c o r r e s p o n d t o t h e a x i s and s u r f a c e t e m p e r a t u r e s r e s p e c t i v e l y 289 XXV LIST OF TABLES T a b l e 5.1 O b s e r v e d ( N g ) and C a l c u l a t e d (N and N ) D i s l o c a t i o n n s i t i e s f o r G a l l i u m A r s e n i a e S i n g l e C r y s t a l s 39 T a b l e 7.1 R e s o l v e d Shear S t r e s s Components i n (001) C r y s t a l s 65 T a b l e 7.2 R e s o l v e d Shear S t r e s s Components i n a (010) P l a n e f o r a [001] C r y s t a l 67 T a b l e 8.1 V a l u e s o f p h y s i c a l p a r a m e t e r s u s e d i n t h e c a l c u l a t i o n s 92 T a b l e 9.1 E f f e c t o f Cone A n g l e on T h e r m a l and S t r e s s F i e l d s 124 T a b l e 9.2 E f f e c t o f Ambient T e m p e r a t u r e on A x i a l T e m p e r a t u r e G r a d i e n t and S t r e s s 150 T a b l e 9.3 E f f e c t o f N o n - l i n e a r i t y o f t h e Ambient T e m p e r a t u r e P r o f i l e on S t r e s s 162 T a b l e 9.4 T a b l e o f Symmetry C o r r e l a t i o n s 198 T a b l e 9.5 E f f e c t o f Thermal C o n d i t i o n s on S t r e s s e s 226 T a b l e VI.1 R o t a t i o n p e r f o r m e d f o r t h e c a l c u l a t i o n o f the RSS 357 T a b l e VI.2 R e s o l v e d Shear S t r e s s component i n t h e <110> {111} s l i p 358 T a b l e VI.3 Compact form of the RSS from T a b l e VI.2 360 XXVI LI S T OF SYMBOLS ( e ) [ B ] * ' M a t r i x o f d e r i v a t i v e s of t h e i n t e r p o l a t i o n f u n c t i o n s i 1 (e ) {F }. I n i t i a l f o r c e v e c t o r at node i i n e l e m e n t e, 2 x 1 . o i 4 x 2 [ B ' ] | e ) N o n - d i m e n s i o n a l [ B ] j e ) m a t r i x , 4 x 2 . [C] C o m p l i a n c e m a t r i x , 4 x 4 . [ C 1 ] N o n - d i m e n s i o n a l [C] m a t r i x , 4 x 4 . ~ ( e ) {d} D i s p l a c e m e n t v e c t o r i n e l e m e n t e, 2 x 1. (e) {d} Nod a l d i s p l a c e m e n t v e c t o r i n element e, 2n x 1. (e ) {d}* N o d a l d i s p l a c e m e n t v e c t o r a t node i e l e m e n t e, 2 x 1 * d * r e d Reduced { d } * e ) v e c t o r , 2 x 1 . {d 1} G l o b a l d i s p l a c e m e n t v e c t o r i n r e d u c e d form, 2n x 1. dV E l e m e n t a l volume. E Young's Modulus, MPa. (e) { F } v F o r c e v e c t o r f o r e l e m e n t e, 2n x 1 . ( e ) {F}. F o r c e v e c t o r at node i i n el e m e n t e, 2 x 1. ( e ) ( e ) {F '} v ; N o n - d i m e n s i o n a l {F }. v e c t o r , 2 x 1 . o o 1 i (e) F 1 E l e m e n t o f {F '} ' v e c t o r , o . o l {F '} N o n - d i m e n s i o n a l g l o b a l f o r c e v e c t o r , 2n x 1. o g Number o f nodes i n e l e m e n t h Heat t r a n s f e r c o e f f i c i e n t ( h . t . c . ) h C o n v e c t i v e h . t . c . c h R a d i a t i v e h . t . c . r (e ) [k] S t i f f n e s s m a t r i x f o r e l e m e n t e, 2n x 2n. (e) [ k ] . . S t i f f n e s s s u b m a t r i x f o r nodes i and j i n e l e m e n t e, i J 2 x 2 . XXV11 [ k ' ] * j * N o n - d i m e n s i o n a l ( k } ! ^ m a t r i x , 2 x 2 . k ^ j E l e m e n t of { M ^ * m a t r i x . [ K 1 ] N o n - d i m e n s i o n a l g l o b a l s t i f f n e s s m a t r i x . (K } F o r c e v e c t o r f o r t e m p e r a t u r e f i e l d c a l c u l a t i o n s . [ K H ] S t i f f n e s s m a t r i x from b o u n d a r y c o n d i t i o n s i n temper- a t u r e f i e l d c a 1 c u a 1 t i o n s . [K^] S t i f f n e s s m a t r i x f o r t e m p e r a t u r e f i e l d c a l c u l a t i o n s L D i f f e r e n t i a l o p e r a t o r . N a t u r a l c o - o r d i n a t e . l j j L e n g t h of e l e m e n t at thne b o u n d a r y . m Degree o f freedom i n e l e m e n t . ->- n U n i t normal v e c t o r t o a s u r f a c e . N T o t a l number of nodes i n the c r y s t a l . [N] I n t e r p o l a t i o n f u n c t i o n s v e c t o r . r R a d i a l c o - o r d i n a t e . r C r i s t a l r a d i u s . o t Time. * t n o n - d i m e n s i o n a l t i m e . T T e m p e r a t u r e , °C. T Q R e f e r e n c e t e m p e r a t u r e . T„_ T e m p e r a t u r e a t m e l t i n g p o i n t . MP Uj R a d i a l d i s p l a c e m e n t of node i . u ( r , z ) R a d i a l d i s p l a c e m e n t a t p o i n t r , z. v Growth v e l o c i t y . Volume o f e l e m e n t e. -(e) Wj A x i a l d i s p l a c e m e n t of node i . w. ( r , z ) A x i a l d i s p l a c e m e n t at p o i n t of r , z. W e i g h t i n g f u n c t i o n . A x i a l c o - o r d i n a t e . C o e f f i c i e n t of t h e r m a l e x p a n s i o n , °C 1 . N o n - d i m e n s i o n a l t e m p e r a t u r e . Ambient n o n - d i m e n s i o n a l t e m p e r a t u r e . S u r f a c e n o n - d i m e n s i o n a l t e m p e r a t u r e . E i g e n v a l u e s of F o u r i e r t y p e a l g e b r a i c e q u a t i o n Shear s t r a i n component i n {£} t e n s o r . A r e a o f element ( c r o s s - s e c t i o n ) . S t r a i n t e n s o r . I n i t i a l s t r a i n t e n s o r . R a d i a l s t r a i n component. A z i m u t h a l s t r a i n component. A x i a l s t r a i n component. N o n - d i m e n s i o n a l a x i a l d i s p l a c e m e n t . E i n g e n v a l u e s of B e s s e l t y p e a l g e b r a i c e q u a t i o n T h e r m a l d i f f u s i v i t y . P o i s s o n ' s r a t i o . N o n - d i m e n s i o n a l r a d i a l d i s p l a c e m e n t . S t r e s s t e n s o r . R a d i a l s t r e s s component. A z i m u t h a l s t r e s s component. A x i a l s t r e s s component. Von M i s e s s t r e s s , MPa. P r i n c i p a l s t r e s s e s . N o n - d i m e n t i o n a l r a d i a l s t r e s s component. N o n - d i m e n t i o n a l a z i m u t h a l s t r e s s component. N o n - d i m e n t i o n a l a x i a l s t r e s s component. S t r e s s tensor at node i , element e. Shear s t r e s s component. Non-dlmentional shear s t r e s s component. XXX ACKNOWLEDGEMENTS I would l i k e t o thank Dr. F r e d W e i n b e r g and I n d i r a V. S a m a r a s e k e r a f o r t h e i r a s s i s t a n c e and v a l u a b l e d i s c u s s i o n s d u r i n g t h e c o u r s e of t h i s work. S u p p o r t form t h e NSERC ( C a n a d a ) , Cominco L t d . , (Canada) and U n i v e r s i d a d N a c i o n a l de M i s i o n e s , ( A r g e n t i n a ) i s g r a t e f u l l y a c k n o w l e d g e d . 1 CHAPTER 1 INTRODUCTION G a l l i u m A r s e n i d e (GaAs) i s a c r y s t a l s e m i c o n d u c t o r compound t h a t i s i n c r e a s i n g l y b e i n g u s e d i n t h e p r o d u c t i o n of s e m i c o n d u c t o r d e v i c e s p a r t i c u l a r l y microwave and m i c r o e l e c t r o n i c . . 1-3 d e v i c e s The a d v a n t a g e s of GaAs o v e r s i l i c o n ( S i ) a r e due t o t h e d i f f e r e n c e s i n p h y s i c a l p r o p e r t i e s between b o t h m a t e r i a l s . E l e c t r o n m o b i l i t i e s i n GaAs a r e about f o u r t i m e s h i g h e r t h a n i n S i w i t h a power c o n s u m p t i o n w h i c h i s a t e n t h o f t h a t i n s i l i c o n . T h ese become p a r t i c u l a r l y s i g n i f i c a n t as t h e VLSI c i r c u i t s become more complex and the o p e r a t i n g s p e e d i n c r e a s e s . In 1974 t h e f i r s t o p e r a t i n g GaAs g a t e was p r o d u c e d . Ten y e a r s l a t e r GaAs IC's have 3 been p r o d u c e d w i t h 10,000 g a t e s or 4 SKRAM . In a d d i t i o n IC's a r e beco m i n g s m a l l e r . A 1.1 by 1.6 mm c h i p may c o n t a i n more t h a n 600 a c t i v e components p l u s c o n t a c t pads, and work a t about 10 4 G b i t / s e c . These f a c t s i n d i c a t e t h a t GaAs i s t h e most p r o m i s i n g 5 m a t e r i a l f o r t h e nex t g e n e r a t i o n o f s u p e r c o m p u t e r s In o p t o e l e c t r o n i c d e v i c e s S i has fewer a p p l i c a t i o n s t h a n GaAs. S i does not glow b r i g h t l y enough and does not l a s e b e c a u s e of t h e band gap s t r u c t u r e On t h e o t h e r hand GaAs LED 1 s a r e b r i g h t , can be made o f d i f f e r e n t c o l o u r s by i m p u r i t y a d d i t i o n s , and GaAs l a s e s w e l l b e c a u s e o f i t s d i r e c t band gap s t r u c t u r e 1 . At p r e s e n t o n l y about 11 * o f the GaAs c h i p s have o p t o e l e c t r o n i c a p p l i c a t i o n s . By 1992 t h e p e r c e n t a g e Is e x p e c t e d t o I n c r e a s e t o 7 25 % . The c o m b i n a t i o n of m i c r o and o p t o e l e c t r o n i c p r o p e r t i e s o f GaAs has l e d to the d e v e l o p m e n t of m o n o l i t h i c o p t o e l e c t r o n i c 8 -10 i n t e g r a t i o n o f s i m p l e c i r c u i t s . S p e c i a l emphasis has been p l a c e d on the d e v e l o p m e n t of the m u l t i q u a n t u m w e l l l a s e r . In t h e a r e a o f s o l a r c e l l s GaAs i s an e x c e l l e n t p h o t o c o n v e r t e r w i t h an e f f i c i e n c y o f 20-24 %, as compared t o S i at 15-18 % . GaAs s o l a r c e l l s a r e e x p e c t e d t o r e a c h e f f i c i e n c i e s o f 30 % w i t h m o n o l i t h i c or h y b r i d tandem c e l l s w h i c h a r e p r e s e n t l y f a b r i c a t e d * * . GaAs d e v i c e s have a d v a n t a g e s i n r a d i a t i o n f i e l d s and a t 4 h i g h e r t e m p e r a t u r e s . They can o p e r a t e i n r a d i a t i o n f i e l d s 10 t i m e s l a r g e r t h a t S i d e v i c e s w i t h o u t d e t e r i o r a t i o n , and a t h i g h e r t e m p e r a t u r e s t h a n s i l i c o n . One o f t h e d i s a d v a n t a g e s of GaAs a t p r e s e n t i s i t s h i g h c o s t 2 2 compared to S i (5.6 cents/mm f o r GaAs compared t o 0.16 cents/mm 1 2 f o r S i ). However a f t e r a l l f a b r i c a t i n g c o s t s a r e c o n s i d e r e d f o r d e v i c e s , the d i f f e r e n c e i n w a f e r c o s t r e p r e s e n t s o n l y a s m a l l f r a c t i o n of t h e t o t a l and i s t h e r e f o r e p r o p o r t i o n a t e l y l e s s 7 s i g n i f i c a n t . One d i s a d v a n t a g e of GaAs, a t p r e s e n t i s i t s h i g h d i s l o c a t i o n d e n s i t y , as compared to s i l i c o n w h ich i s d i s l o c a t i o n f r e e . D i s l o c a t i o n i n many c a s e s may a f f e c t b o t h t h e y i e l d o f d e v i c e s d u r i n g f a b r i c a t i o n and t h e p e r f o r m a n c e o f t h e d e v i c e s . 3 T h i s i n v e s t i g a t i o n f o c u s e s on t h e d i s l o c a t i o n s p r e s e n t i n m e l t grown GaAs. The d i s l o c a t i o n s a r e b e l i e v e d t o be g e n e r a t e d d u r i n g l i q u i d e n c a p s u l a t e d C z o c h r a l s k i growth (LEC) by t h e r m a l s t r e s s e s i n t h e c r y s t a l . A m a t h e m a t i c a l model w i l l be d e v e l o p e d to d e t e r m i n e the t h e r m a l and s t r e s s f i e l d s i n the g r o w i n g c r y s t a l as a f u n c t i o n of t h e many growth v a r i a b l e s . The o b j e c t i v e i s t h e n t o use t h i s model to c o n t r o l and m o d i f y the g r o w th c o n d i t i o n s and t h u s r e d u c e t h e d i s l o c a t i o n d e n s i t y i n a c o n t r o l l e d way. 4 CHAPTER 2 THE GROWTH OF BULK GaAs CRYSTALS 2 . 1 The L i q u i d E n c a p s u l a t e d C z o c h r a l s k i T e c h n q u e (LEC) 13-15 I n t h e LEC t e c h n i q u e , shown s c h e m a t i c a l l y i n F i g u r e 2.1, a s i n g l e c r y s t a l o f c o n t r o l l e d o r i e n t a t i o n i s p u l l e d f r o m t h e m e l t , v e r t i c a l l y u p w a r d s , s t a r t i n g f r o m a s e e d c r y s t a l . B o t h t h e s e e d and t h e m e l t a r e r o t a t e d a t d i f f e r e n t v e l o c i t i e s and i n d i r e c t i o n s c o u n t e r t o e a c h o t h e r . I n g e n e r a l h i g h p u r i t y Ga and As a r e p l a c e d i n t h e m e l t c r u c i b l e and t h e i n t e r m e t a l 1 i c GaAs s y n t h e s i z e d i n t h e h i g h p r e s s u r e c r y s t a l p u l l e r . Ga m e l t s a t 20.8°C. As m e l t s a t 817°C a t 28 a t m o s p h e r e s . The m e l t i n g t e m p e r a t u r e o f GaAs i s 1238°C. A t t h i s t e m p e r a t u r e t h e a r s e n i c p a r t i a l p r e s s u r e i s 0.98 atm. P r e s s u r e s o f 50 atm. a r e t h e r e f o r e r e q u i r e d t o s y n t h e s i z e GaAs. Once t h e GaAs has f o r m e d , a much l o w e r p r e s s u r e o f s e v e r a l a t m o s p h e r e s may be u s e d t o k e e p t h e GaAs f r o m d e c o m p o s i n g . C o m m e r c i a l c r y s t a l g r o w e r s , i n p a r t i c u l a r t h e " M e l b o u r n " g r o w e r f r o m C a m b r i d g e I n s t r u m e n t s L t d . i s u s e d e x t e n s i v e l y t o grow GaAs c r y s t a l s . P r e s s u r e s i n s i d e t h e chamber can be r a i s e d t o 50-70 atm. as r e q u i r e d . I n a d d i t i o n t o t h e p r e s s u r e i n s i d e t h e g r o w i n g chamber, i t was f o u n d t h a t l i q u i d e n c a p s u l a t i o n o f t h e m e l t and t h e s o l i d a t t h e h i g h e s t t e m p e r a t u r e was n e c e s s a r y t o m a i n t a i n s t o i c h i o m e t r y . T h i s i s done by e n c a p s u l a t i n g t h e l i q u i d w i t h B 0_ w h i c h a l s o c o v e r s t h e l o w e r 2 o end o f t h e g r o w i n g c r y s t a l . The m e l t and e n c a p s u l a n t a r e le F i g u r e 2.1 Schemat i c o f a LEC p u l l i n g chamber 1 fi c o n t a i n e d i n a q u a r t z or a p y r o l i t i c b o r o n n i t r i d e c r u c i b l e . U s i n g t h e h i g h p r e s s u r e (HP) LEC p r o c e s s i n t h e M e l b o u r n p u l l e r , c o n t r o l l e d o r i e n t a t i o n <100> c r y s t a l s up t o 75 mm i n d i a m e t e r and 8 kgm i n w e i g h t can be grown w i t h d i s l o c a t i o n d e n s i t i e s i n the o r d e r o f 1 0 4 / cm 2 As an a l t e r n a t i v e t o t h e h i g h p r e s s u r e c r y s t a l g r o w e r s , s i n g l e c r y s t a l s a r e b e i n g s u c c e s s f u l l y grown a t 1-2 a t m o s p h e r e s p r e s s u r e u s i n g t h e LEC t e c h n i q u e ( L P L E C ) . In t h i s c a s e t h e s t a r t i n g m a t e r i a l i s p o l y c r y s t a l 1 i n e GaAs w h i c h has been s y n t h e s i z e d s e p a r a t e l y i n a p r e s s u r e v e s s e l . A n o v e l 4 17 t e c h n i q u e ' p e r m i t s i n - s i t u s y n t h e s i s i n t h e LP gro w e r s by i n j e c t i o n o f As t h r o u g h t h e B O l a y e r i n t o a Ga m e l t c o n t a i n e d i n the c r u c i b l e . With the LPLEC g r o w e r s , c r y s t a l s of 100 mm 18 — 20 d i a m e t e r and 14 kgm w e i g h t have been p r o d u c e d The c r y s t a l growth p r o c e s s s t a r t s w i t h a s e e d c r y s t a l , u s u a l l y o r i e n t e d f o r <100> or <111> grow t h , w h i c h i s d i p p e d i n t o t h e m e l t . The c r y s t a l i s t h e n n e c k e d t o r e d u c e t h e p r o p a g a t i o n o f 2 1 d i s l o c a t i o n s from t h e seed t o t h e c r y s t a l and t h e d i a m e t e r t h e n s l o w l y i n c r e a s e d t o t h e f u l l d i a m e t e r o f t h e c r y s t a l . The growth i s m o n i t o r e d v i s u a l l y and w i t h a TV camera. Once the d i a m e t e r of the c r y s t a l i s e s t a b l i s h e d by manual c o n t r o l o f the power i n p u t , f u r t h e r c o n t r o l i s done w i t h an a u t o m a t i c w e i g h i n g s y s t e m w h i c h a d j u s t s t h e power i n p u t w i t h computer 2 2-24 c o n t r o l Human e r r o r s , i n a b i l i t i e s t o see c l e a r l y t h r o u g h the e n c a p s u l a n t , and s y s t e m a t i c e r r o r s i n the w e i g h i n g p r o c e d u r e 2 5 due t o c a p i l l a r y f o r c e s , r e s u l t i n v a r i a t i o n s i n t h e c r y s t a l 7 d i a m e t e r o f t h e o r d e r o f 5 mm. A t t e m p t s a r e b e i n g made t o r e d u c e t h e s e p r o b l e m s 6 ' 1 9 ' 2 6 . When c r y s t a l g r o w t h - i s c o m p l e t e d , the c r y s t a l i s removed from t h e ^z°3 a n d s l o w l y c o o l e d i n the chamber t o m i n i m i z e t h e r m a l s t r e s s e s . GaAs i s n o r m a l l y s e m i - i n s u l a t i n g . In some c a s e s i soe 1 ec t r on i c a d d i t i o n s , s u c h as In, a r e added t o the m e l t t o improve p r o p e r t i e s . O t h e r s o l u t e a d d i t i o n s a r e made t o t h e m e l t t o make t h e m a t e r i a l n- or p - t y p e s e m i c o n d u c t o r s . F o r n - t y p e m a t e r i a l d o p a n t s s u c h as S, Se, Te, Sn, S i and Ge 2 7 a r e added. F o r p - t y p e m a t e r i a l Zn and Mg a r e added. The e f f e c t o f s o l u t e a d d i t i o n s and i m p u r i t i e s on t h e e l e c t r i c a l and m e c h a n i c a l p r o p e r t i e s of LEC-grown GaAs c r y s t a l s and d e v i c e s a r e d i s c u s s e d below. 2.2 C l a s s i f i c a t i o n of D e f e c t s i n LEC grown GaAs c r y s t a l A d e f e c t i s d e f i n e d as any d e v i a t i o n from t h e c r y s t a l s t r u c t u r e . D e f e c t s c o n s i s t of : i . D i s l o c a t i o n s i i . C h e m i c a l i m p u r i t i e s i i i . D e v i a t i o n from s t o i c h i o m e t r y i v . L a t t i c e d e f e c t s , 1. v a c a n c i e s and i n t e r s t i t i a l s ; 2. s u b g r a i n s , g r a i n b o u n d a r i e s , s t a c k i n g f a u l t s , s t r i a t i o n s and o t h e r s ; 8 3. i n c l u s i o n s and p r e c i p i t a t e s ; 4. m i c r o d e f e c t s , c o n s i s t i n g of s m a l l d i s l o c a t i o n l o o p s , p r e c i p i t a t e s , h e l i c o i d a l d i s l o c a t i o n s and o t h e r s . These d e f e c t s have c h a r a c t e r i s t i c l e n g t h s o f 100 t o 1 n m 2 8 " 3 0 . The o r i g i n and n a t u r e o f many o f t h e s e d e f e c t s i s not 30 31 c l e a r l y u n d e r s t o o d ' . G e n e r a l l y a number of d e f e c t s a r e p r e s e n t e d i n t h e c r y s t a l , w h i c h i n t e r a c t w i t h e a c h o t h e r and m u l t i p l y . An example i s t h e p r e s e n c e of s m a l l p r e c i p i t a t e s on 3 2 d i s l o c a t i o n l i n e s . The p r e c i p i t a t e s a r e b e l i e v e d t o be c r y s t a l l i t e s o f As embedded i n Ga m a t r i c e s r e s u l t i n g from 1) d i r e c t c o n d e n s a t i o n of As p o i n t d e f e c t s or 2) s w e e p i n g up o f e x i s t i n g p r e c i p i t a t e s by moving d i s l o c a t i o n s . 2 . 3 D i s l o c a t i o n s i n LEC-GaAs L a r g e d i a m e t e r GaAs <100> s i n g l e c r y s t a l s p r o d u c e d by a 4 M e l b o u r n p u l l e r have a d i s l o c a t i o n d e n s i t y of a p p r o x i m a t e l y 5x10 2 3 3 /cm . C o n s i d e r a b l e e f f o r t i s b e i n g d i r e c t e d t o w a r d p r o d u c i n g low o r z e r o d e n s i t y l a r g e d i a m e t e r GaAs c r y s t a l s . In 1982 t h e l a r g e s t undoped GaAs c r y s t a l grown f r e e o f d i s l o c a t i o n was 15 mm34, i n 1985 a c r y s t a l 50mm i n d i a m e t e r was r e p o r t e d t o have been grown d i s l o c a t i o n and s t r i a t i o n f r e e ' . In t h e l a t t e r c a s e t h e c r y s t a l was grown u s i n g a m o d i f i e d LEC t e c h n i q u e c a l l e d VMFEC wh i c h i n c l u d e s a v e r t i c a l m a g n e t i c f i e l d . The c r y s t a l i n t h i s c a s e i s f u l l y e n c a p s u l a t e d and the m e l t i s doped. The d i s l o c a t i o n d e n s i t y can be r e d u c e d t o 10° /cm by d o p i n g t h e m e l t w i t h n, p or i s o e 1 e c t r o n i c I m p u r i t i e s . R e c e n t improvements o f the t h e r m a l c o n d i t i o n s d u r i n g growth have r e s u l t e d i n undoped 3 2 c r y s t a l s w i t h e t c h p i t d e n s i t i e s (EPD) of 5x10 /cm f o r 5 cm 3 7-39 d i a m e t e r c r y s t a l . In one c a s e 70 mm <100> undoped c r y s t a l s were grown i n low v e r t i c a l t e m p e r a t u r e g r a d i e n t s h a v i n g EPD of 3 2 5x10 /cm o v e r 70 % of t h e w a f e r a r e a and t h r o u g h o u t 75 % o f t h e 4 0 i n g o t l e n g t h T y p i c a l d i s l o c a t i o n d i s t r i b u t i o n s on t r a n s v e r s e s e c t i o n s of GaAs c r y s t a l a r e shown i n F i g u r e 2.2. In F i g u r e 2 . 2 a 4 1 the d i s l o c a t i o n d e n s i t y (EDP) i s shown a l o n g <100> and <110> d i r e c t i o n s on a (100) GaAs w a f e r s u r f a c e s h o w i n g maximum d i s l o c a t i o n s a t the o u t s i d e and minimum midway between the o u t s i d e and t h e c e n t e r , g i v i n g a W shape d i s t r i b u t i o n a c r o s s t h e e n t i r e w a f e r . F i g u r e 2.2b shows EPD d i s t r i b u t i o n i n a (100) w a f e r s u r f a c e t a k e n n e a r th e t a i l end of a 5 cm d i a m e t e r c r y s t a l . The e t c h p i t d i s t r i b u t i o n has the f o u r - f o l d symmetry n o r m a l l y o b s e r v e d , w i t h h i g h e r d e n s i t i e s at t h e o u t s i d e and c e n t r e o f t h e w a f e r . C l o s e t o the end o f the c r y s t a l , EPD maps on 5.0 cm d i a m e t e r c r y s t a l show the f o u r - f o l d symmetry d e g e n e r a t e s t o two- f o l d symmetry. T h i s d i s t o r t i o n i s c a u s e d by the s h i f t o f t h e two minima i n the [110] and [110] d i r e c t i o n s to p o s i t i o n s c l o s e r t o 3 3 t h e edge o f t h e w a f e r 10 C I 1 0 ] C 1003 b 2.2 a) E x p e r i m e n t a l l y d e t e r m i n e d 4 1 d i s l o c a t i o n d e n s i t y a l o n g [ 1 1 0 ] and [ 0 1 0 ] . b) EPD m a p 3 3 on a 5cm d i a . w a f e r f r o m t h e t a i l end . 1 0 4 E t c h P i t s / c m . 1 < y e l l o w < 12 < g r e e n < 15 < b l u e < 20 < b l a c k < 30 . 11 2. 4 E f f e c t o f D i s l o c a t i o n s on P r o p e r t i e s o f GaAs D e v i c e s D i s l o c a t i o n s a r e known t o a f f e c t t h e e l e c t r o n i c p r o p e r t i e s o f GaAs i n t h e f o l l o w i n g ways 1) They a r e c o n s i d e r e d t o be 4 2 n o n - r a d i a t i v e r e c o m b i n a t i o n c e n t e r s ; 2) They a b s o r b non A Q r a d i a t i v e i m p u r i t y atoms or p o i n t d e f e c t s ; 3) When d i s l o c a t i o n s a r e a r r a y e d i n c e l l u l a r s t r u c t u r e s , c a r r i e r s a r e not 4 3 a c t i v e at t h e c e l l w a l l s The e f f e c t of d i s l o c a t i o n s on t h e p r o p e r t i e s o f m i c r o e l e c t r o n i c and o p t o e l e c t r o n i c d e v i c e s depends on t h e n a t u r e o f t h e d e v i c e and how i t i s f a b r i c a t e d . In o p t o e l e c t r o n i c d e v i c e s , i t has been r e p o r t e d t h a t d i s l o c a t i o n s s e r i o u s l y r e d u c e 4 4 4 2 t h e l i f e t i m e o f l a s e r s , r e d u c e the e f f i c i e n c y o f L E D 1 s , and under an a p p l i e d s t r e s s , i n c r e a s e t h e d e g r a d a t i o n o f d i o d e s by one o r d e r of m a g n i t u d e 4 5 , 4 6 . The mechanism f o r t h e s e e f f e c t s i s 47 u n c l e a r W i t h m i c r o e l e c t r o n i c d e v i c e s i t was c o n s i d e r e d t h a t A Q d i s l o c a t i o n s d i d not s i g n i f i c a n t l y a f f e c t t h e i r p r o p e r t i e s However a t the p r e s e n t time t h i s has changed and d i s l o c a t i o n s a r e b e l i e v e d t o a f f e c t t h e e l e c t r i c a l p r o p e r t i e s o f IC's o f GaAs. These r e s u l t s a r e p r e s e n t e d and d i s c u s s e d i n A p p e n d i x I. 12 CHAPTER 3 THE ORIGIN OF DISLOCATIONS IN LEC GaAs CRYSTALS D i s l o c a t i o n s i n GaAs a r e e i t h e r s e e d r e l a t e d or growth r e l a t e d . The s e e d r e l a t e d d i s l o c a t i o n s r e s u l t from t h e p r o p a g a t i o n and m u l t i p l i c a t i o n o f d i s l o c a t i o n s from t h e s e e d i n t o t h e main c r y s t a l . These d i s l o c a t i o n s can be r e d u c e d or e l i m i n a t e d by n e c k i n g the s e e d c r y s t a l , as i s commonly done, or by s t a r t i n g 3 5 w i t h a d i s l o c a t i o n f r e e s e e d c r y s t a l The growth r e l a t e d d i s l o c a t i o n s i n t h e c r y s t a l a r e g e n e r a l l y a t t r i b u t e d t o t h e t h e r m a l s t r e s s e s g e n e r a t e d i n t h e c r y s t a l d u r i n g g rowth. The d i s l o c a t i o n s n u c l e a t e i n r e g i o n s of maximum s t r e s s n e a r t h e c r y s t a l s u r f a c e , t h e n p r o p a g a t e a l o n g s l i p p l a n e s and m u l t i p l y . The d i s l o c a t i o n s can a l s o be g e n e r a t e d by t h e r m a l s t r e s s e s a t p r e c i p i t a t e s or f o r e i g n p a r t i c l e s , and by t h e c o n d e n s a t i o n o f v a c a n c i e s i n t o d i s l o c a t i o n l o o p s w h i c h g l i d e . E x a m i n a t i o n of a <100> me l t grown GaAs° c r y s t a l w a f e r by t r a n s m i s s i o n X - r a y t o p o g r a p h y o f e t c h p i t s (EPD) shows t h a t t h e d i s l o c a t i o n s a r e not u n i f o r m l y d i s t r i b u t e d t h r o u g h o u t t h e w a f e r as shown i n F i g u r e s 3.1 a a t C . As d e s c r i b e d p r e v i o u s l y the d i s l o c a t i o n d e n s i t y a l o n g a c r y s t a l d i a m e t e r f o l l o w s a W p a t t e r n w i t h h i g h e r c o n c e n t r a t i o n s a t the o u t s i d e and c e n t e r . In a d d i t i o n t h e d i s l o c a t i o n s form c e l l s by c o n c e n t r a t i n g a l o n g c e l l w a l l s . C e l l d i a m e t e r s have been o b s e r v e d o f 0.1 mm i n a r e a s of 13 F i g u r e 3.1 T r a n s m i s s i o n X - r a y t o p o g r a p h y i n GaAs - ( a ^ ^ a n d (b) undoped ; (c) Te-doped, (110) a x i a l s e c t i o n 14 5 2 d i s l o c a t i o n d e n s i t i e s o f 10 /cm ( b a s e d on EPD). At l o w e r d e n s i t i e s t h e c e l l s can i n c r e a s e i n s i z e up t o 3 m m 3 0 , 1 1 0. The c e l l w a l l d i r e c t i o n s on a (100) c r y s t a l f a c e a r e r a n d o m l y o r i e n t e d . The f o r m a t i o n of c e l l s i n t h e GaAs w a f e r s i s not c l e a r l y u n d e r s t o o d . The most f a v o u r e d mechanism i s t h a t d i s l o c a t i o n s g e n e r a t e d n e a r t h e c r y s t a l w a l l s and i n t h e c e n t r e g l i d e and c l i m b a f t e r s o l i d i f i c a t i o n e v e n t u a l l y f o r m i n g t h e c e l l s t r u c t u r e t o m i n i m i z e t h e i r e n e r g y . An a l t e r n a t i v e mechanism has been p r o p o s e d by H o l m e s 1 1 1 i n w h i c h t h e c e l l s t r u c t u r e i s a t t r i b u t e d to c o n s t i t u t i o n a l s u p e r c o o l i n g a t the s u r f a c e , s i m i l a r to c e l l 112 f o r m a t i o n i n m e t a l a l l o y s . T h i s mechanism a p p e a r s u n l i k e l y b e c a u s e o f the low i m p u r i t y l e v e l s i n the GaAs, h i g h s t o i c h i o m e t r y a t t h e i n t e r f a c e and slow g r o w th v e l o c i t i e s . In a d d i t i o n to c e l l s t r u c t u r e s on GaAs c r y s t a l s , l o n g l i n e a r a r r a y s o f d i s l o c a t i o n s a r e a l s o o b s e r v e d as shown i n F i g u r e 3.1b at L. The l i n e a r a r r a y s a r e o b s e r v e d i n a r e a s h a v i n g a 4 2 d i s l o c a t i o n d e n s i t y of l e s s t h a n 2x10 /cm and p r o d u c e a s m a l l t i l t o r i e n t a t i o n d i f f e r e n c e o f about 0.01° i n t h e c r y s t a l a c r o s s t h e l i n e . These a r r a y s a r e r o u g h l y s i m i l a r to l i n e a g e sub- b o u n d a r i e s i n m e l t grown m e t a l c r y s t a l s a t t r i b u t e d t o t h e c o n d e n s a t i o n of v a c a n c i e s i m m e d i a t e l y b e h i n d t h e i n t e r f a c e , f o r m i n g d i s l o c a t i o n l o o p s w h i c h c l i m b to form l i n e a g e b o u n d a r i e s . S l i p l i n e s a r e sometimes o b s e r v e d on e t c h e d GaAs w a f e r s as shown i n F i g u r e 3.1a a t S, g e n e r a l l y i n a r e a s o f l o w e r d i s l o c a t i o n d e n s i t y ( 1 0 3 / c m 2 ) 1 1 3 . 15 S p e c i f i c o b s e r v a t i o n s and d i s c u s s i o n s of d i s l o c a t i o n g e n e r a t i o n and m u l t i p l i c a t i o n i n GaAs a r e g i v e n below : 1. The g e n e r a t i o n o f d i s l o c a t i o n s by t h e r m a l s t r e s s e s has 114 been r e p o r t e d by M i l v i d s k i i and c o - w o r k e r s and o t h e r s 1 1 ^ * ' 1 1 * 5 . T h e r m a l s t r e s s e s i n t h e g r o w i n g c r y s t a l r e s u l t from r a d i a l and a x i a l t e m p e r a t u r e g r a d i e n t s . When t h e l o c a l s t r e s s i s g r e a t e r t h a n t h e c r i t i c a l r e s o l v e d s h e a r s t r e s s , g l i d e o c c u r s on t h e _ 117 118 (111) s l i p p l a n e s i n <110> d i r e c t i o n s ' . T h i s 119 mechanism was p r o p o s e d f o r S i by B i l l i g and was 12 0 a p p l i e d t o Ge by P e n n i n g 2. The most p r o b a b l e mechanism f o r t h e g e n e r a t i o n o f d i s l o c a t i o n s i s s t r e s s c o n c e n t r a t i o n a t h e t e r o g e n e o u s s o u r c e s at the c r y s t a l s u r f a c e or i n t h e b u l k m a t e r i a l . S m a l l p r e c i p i t a t e s o f As have been f o u n d a t 3 2 d i s l o c a t i o n s Amorphous p r e c i p i t a t e s o f Ga or As have 3 0 been o b s e r v e d on the c e l l w a l l s i n undoped GaAs I n s u l a r p o l y c r y s t a l 1 i n e GaAs h a v i n g d i a m e t e r s o f 1000°A have been o b s e r v e d i n S i doped GaAs w h i c h a r e 30 a s s o c i a t e d w i t h d i s l o c a t i o n g e n e r a t i o n . S t e i n e m a n n 121 and Z i m merly a s s o c i a t e d i s l o c a t i o n g e n e r a t i o n w i t h p r e c i p i t a t e s of e x c e s s Ge or i m p u r i t y atoms. E x c e s s Ga v a c a n c i e s 1 1 ^ ' 1 2 2 i n p a r t i c u l a r and n a t i v e p o i n t d e f e c t s 123125 i n g e n e r a l w h i c h p r e c i p i t a t e i n t o d i s l o c a t i o n l o o p s have been c o n s i d e r e d as a s o u r c e of d i s l o c a t i o n s . 12 6 12 7 L a g o w s k i et a l . ' have p r o p o s e d a t h e o r y i n w h i c h 16 t h e Ga v a c a n c y Is t h e c r i t i c a l n a t i v e d e f e c t i n the f o r m a t i o n of t h e d i s l o c a t i o n l o o p s . T h i s t h e o r y i s p r e s e n t e d i n p a r t b o f A p p e n d i x I. o 3. The s t r e s s d i s t r i b u t i o n i n g r o w i n g c r y s t a l s has been 12 8 c o n s i d e r e d i n s e v e r a l ways. N i k i t e n k o and Indenbom used p h o t o e l a s t i c methods t o o b s e r v e the s t r e s s d i s t r i b u t i o n i n c r y s t a l s . They a l s o used m a t h e m a t i c a l models t o e s t a b l i s h t h e s t r e s s f i e l d i n a g r o w i n g c r y s t a l and d e r i v e d t h e d i s l o c a t i o n d i s t r i b u t i o n b a s e d on l o c a l s t r e s s l e v e l s above the c r i t i c a l r e s o l v e d s h e a r s t r e s s . The d i s t r i b u t i o n t h e y p r e d i c t e d a g r e e d r e a s o n a b l y w e l l w i t h o b s e r v a t i o n s . The c o r r e l a t i o n of t h e r m a l s t r e s s f i e l d s t o d i s l o c a t i o n s has a l s o been o b s e r v e d u s i n g X - r a y d i f f r a c t i o n to measure l a t t i c e s t r a i n s i n G a A s 1 2 9 - 1 3 1 , p h o t o e l a s t i c m e t h o d s 1 3 2 , 1 3 3 and p h o t o 1 u m i n e s c e n t s c a n n i n g 1 3 4 . In summary, s e v e r a l c a u s e s f o r t h e o r i g i n o f d i s l o c a t i o n s i n GaAs have been p r o p o s e d i n t h e l i t e r a t u r e . In c r y s t a l s grown from m e l t s f a r from s t o i c h i o m e t r y t h e Ga v a c a n c y may c o n t r o l t h e f o r m a t i o n o f d i s l o c a t i o n s . Under normal growth c o n d i t i o n s , however s t r e s s e s p l a y a f u n d a m e n t a l r o l e i n c o n t r o l l i n g d i s l o c a t i o n d e n s i t i e s by a g l i d e mechanism. More e x p e r i m e n t a l e v i d e n c e s u p p o r t i n g the s t r e s s r e l a t e d mechanism i s p r e s e n t e d i n t h e n e x t c h a p t e r . 17 CHAPTER 4 DISLOCATION DENSITY AND CRYSTAL GROWTH The d e n s i t y and d i s t r i b u t i o n o f d i s l o c a t i o n s i n LEC grown GaAs c r y s t a l s i s s t r o n g l y d e p e n d e n t on a number o f g r o w t h p a r a m e t e r s . The d i s l o c a t i o n d e n s i t y may be r e d u c e d , i m p r o v i n g t h e c r y s t a l q u a l i t y , by c h a n g i n g t h e g r o w t h c o n d i t i o n s : 1) R e d u c i n g t h e r a d i a l and a x i a l t h e r m a l g r a d i e n t s d u r i n g g r o w t h 2) A d d i n g i m p u r i t i e s t o s o l u t i o n h a r d e n t h e c r y s t a l i n c r e a s i n g the CRSS 3) R e d u c i n g v a r i a t i o n s i n t h e c r y s t a l r a d i u s d u r i n g g r o w t h 4) M a i n t a i n i n g s t o i c h i o m e t r y o f t h e Ga and As i n t h e m e l t . More u n i f o r m dopant and i m p u r i t y d i s t r i b u t i o n i n t h e c r y s t a l may be o b t a i n e d by s t o p p i n g f l u i d f l o w i n t h e m e l t due to c o n v e c t i on. 4•1 S t r e s s e s i n t h e C r y s t a l Due t o T h e r m a l G r a d i e n t s T h e r m a l s t r e s s e s i n the c r y s t a l i n LEC GaAs growth depend p r i m a r i l y on 1) t h e c r y s t a l d i m e n s i o n s and 2) t h e ambient v a r i a b l e s a f f e c t i n g t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e c r y s t a l . 18 4.1.1 C r y s t a l D i m e n s i o n s In t h i s c a t e g o r y t h e v a r i a b l e s a r e c r y s t a l d i a m e t e r , c r y s t a l l e n g t h , cone a n g l e and neck and s e e d d i m e n s i o n s . I n c r e a s i n g t h e d i a m e t e r of t h e c r y s t a l g e n e r a l l y i n c r e a s e s t h e d i s l o c a t i o n d e n s i t y . At v e r y l a r g e d i a m e t e r s t h i s may not be t h e c a s e . I n c r e a s i n g t h e d i a m e t e r o f c r y s t a l s from 5.0 - 7.5 cm d i a m e t e r to 12.7 - 15.2 cm d i a m e t e r d i d n o t change the 1 8 d i s l o c a t i o n d e n s i t y The i n c r e a s e i n d i s l o c a t i o n d e n s i t y w i t h i n c r e a s i n g d i a m e t e r can be due t o an i n c r e a s e i n a x i a l a n d / o r r a d i a l t h e r m a l 13 5 13 6 g r a d i e n t s . E x p e r i m e n t a l o b s e r v a t i o n s a r e c o n t r a d i c t o r y R e s u l t s o f c a l c u l a t i o n s a r e a l s o c o n t r a d i c t o r y . A c c o r d i n g t o 13 7 J o r d a n e t a l . b o t h a x i a l and r a d i a l g r a d i e n t s i n c r e a s e w i t h 13 5 r a d i u s . On t h e o t h e r hand, B r i c e and B u c k l e y - G o l d e n and 13 8 Humphreys r e p o r t t h a t a x i a l t e m p e r a t u r e g r a d i e n t s and a x i a l t e m p e r a t u r e c u r v a t u r e s d e c r e a s e w i t h i n c r e a s i n g r a d i i . The EPD i n d i c a t i n g d i s l o c a t i o n d e n s i t y , i n c r e a s e s w i t h d i s t a n c e from t h e s e e d end o f t h e c r y s t a l . The d i s l o c a t i o n d i s t r i b u t i o n a c r o s s t h e c r y s t a l has a W shape, as shown i n F i g u r e 4.1., w h i c h i s m a i n t a i n e d t h r o u g h o u t the c r y s t a l . In low 113 d i s l o c a t i o n d e n s i t y c r y s t a l s E l l i o t e t a l . have r e p o r t e d h i g h EPD v a l u e s a t b o t h t h e s e e d and t a i l ends o f t h e c r y s t a l . The e f f e c t of cone a n g l e on EPD has been e v a l u a t e d e x p e r i m e n t a l l y by Chen and H o l m e s 1 1 0 . They f o u n d t h a t t h e EPD d e c r e a s e s w i t h i n c r e a s i n g cone a n g l e up t o a v a l u e o f a b o u t 19 F i g u r e 4.1 R a d i a l d i s l o c a t i o n d e n s i t y p r o f i l e s a c r o s s w a f e r s o b t a i n e d from t h e f r o n t , m i d d l e , and t a i l o f a c r y s t a l . The r a d i a l p r o f i l e s a r e "W" shaped, and t h e a v e r a g e EPD i n c r e a s e s from t h e f r o n t t o t h e t a i l . 20 25 d e g r e e s . Above t h i s v a l u e t h e y f o u n d no c o r r e l a t i o n between 3 9 t h e EPD and cone a n g l e . T h i s a g r e e s w i t h Watanabe et a l . T h e r m a l s t r e s s e s depend on t h e e l a s t i c c o n s t a n t s of GaAs. A c c o r d i n g l y t h e c r y s t a l o r i e n t a t i o n s h o u l d a f f e c t t h e d i s l o c a t i o n 1 1 R d e n s i t y . F o r h o r i z o n t a l Bridgman g r o w t h o f GaAs P l a s k e t e t a l . have f o u n d t h a t the l o w e s t d i s l o c a t i o n d e n s i t y was o b t a i n e d f o r c r y s t a l s grown i n t h e <013> d i r e c t i o n . T h i s c a n n o t be t a k e n a d v a n t a g e of b e c a u s e c i r c u l a r GaAs c r y s t a l w a f e r s h a v i n g (100) s u r f a c e s a r e r e q u i r e d f o r p r o d u c t i o n o f I C ' s . 4.1.2 T h e r m a l C o n d i t i o n s D u r i n g C r y s t a l Growth T h e r m a l g r a d i e n t s i n t h e c r y s t a l depend d i r e c t l y on t h e t h e r m a l c o n d i t i o n s i n t h e s u r r o u n d i n g e n v i r o n m e n t t h r o u g h h e a t t r a n s f e r at t h e b oundary of t h e c r y s t a l . The h e a t t r a n s f e r p r o c e s s can be c h a r a c t e r i z e d i n a s i m p l e way by h e a t t r a n s f e r c o e f f i c i e n t s and t h e ambient t e m p e r a t u r e a r o u n d th e c r y s t a l . The main f a c t o r s a f f e c t i n g the h e a t t r a n s f e r a r e : gas p r e s s u r e , b o r o n o x i d e l a y e r t h i c k n e s s , and t h e geometry and p o s i t i o n o f t h e h e a t e r and s h i e l d i n g s y s t e m . I n c r e a s i n g gas p r e s s u r e i n c r e a s e s the E P D 3 9 ' 1 1 0 ' 1 1 ^ . T h i s e f f e c t i s e x p l a i n e d i n terms o f t h e v a r i a t i o n o f t h e h e a t t r a n s f e r c o e f f i c i e n t 1 1 0 , w h i c h a c c o r d i n g t o c a l c u l a t i o n s 13 7 13 9 i n c r e a s e s as the s q u a r e r o o t o f t h e p r e s s u r e ' The change i n h e a t t r a n s f e r c o e f f i c i e n t between the c r y s t a l and gas c a n n o t e x p l a i n t h e v a r i a t i o n i n t h e EPD o b s e r v e d . 21 The gas p r e s s u r e d u r i n g c r y s t a l growth c a n n o t be r e d u c e d to r e d u c e d i s l o c a t i o n d e n s i t y . W i t h o u t s u f f i c i e n t p r e s s u r e at h i g h t e m p e r a t u r e s As atoms l e a v e t h e c r y s t a l . E x c e s s Ga on t h e s u r f a c e w i l l r e s u l t i n p i t t e d s u r f a c e s , t w i n s and p o l y c r y s t a l i n i t y 110,140,141 j n a d d i t i o n , c h a n g i n g t h e gas p r e s s u r e changes th e a t o m i c f r a c t i o n o f As i n t h e m e l t . T h i s can i n f l u e n c e the 14 2 e l e c t r i c a l r e s i s t i v i t y o f t h e m a t e r i a l The d i s l o c a t i o n d e n s i t y i s a l s o a f f e c t e d by t h e t y p e of i n e r t gas u s e d . A n i t r o g e n a t m o sphere g i v e s an EPD one o r d e r of 3 9 m agnitude h i g h e r t h a n a r g o n . W i t h o t h e r i n e r t a t m o s p h e r e s t h e EPD c o u n t s were f o u n d t o i n c r e a s e i n the o r d e r K r , A r , and 143 144 He ' The e f f e c t i s a t t r i b u t e d t o t h e d i f f e r e n t h e a t t r a n s f e r e f f i c i e n c i e s o f the g a s e s which change th e t h e r m a l g r a d i e n t i n the b o r o n o x i d e e n c a p s u l a n t . In a c o n v e n t i o n a l M e l b o u r n grower i n c r e a s i n g t h e B O l a y e r d e c r e a s e s the d i s l o c a t i o n d e n s i t y i n the c r y s t a l . D o u b l i n g t h e e n c a p s u l a n t t h i c k n e s s from 15 t o 30 mm r e d u c e s by h a l f t h e a x i a l , . ,.145-147 . . * c « , J 3 5 , 1 4 8 g r a d i e n t . At p r e s e n t 50 mm t h i c k l a y e r s a r e u s e d Enhanced h e a t i n g o f the B 2 ° 3 l a y e r t h r o u g h windows i n t h e s u s c e p t o r c y l i n d e r r e s u l t s i n lower a x i a l g r a d i e n t s and t h e r e f o r e l o w e r EPD. The a d d i t i o n of b a f f l e s above the c r u c i b l e t o r e d u c e c o n v e c t i o n of t h e gas r e d u c e s the a x i a l and r a d i a l g r a d i e n t s t o 15°C/cm and 3°C/cm r e s p e c t i v e l y f o r a B o 0 - t h i c k n e s s o f 30 mm. With t h i s t e c h n i q u e 5 cm d i a m e t e r c r y s t a l s can be p r o d u c e d w i t h 3 2 d i s l o c a t i o n d e n s i t i e s of 10 /cm i n l a r g e a r e a s . 22 4.2 A l l o y A d d i t i o n to the GaAs The r e d u c t i o n or e l i m i n a t i o n o f d i s l o c a t i o n s i n GaAs by 14 9 a l l o y a d d i t i o n s to the m e l t was r e p o r t e d as e a r l y as 1972 . The e f f e c t i v e n e s s o f s p e c i f i c a d d i t i o n s i n r e d u c i n g the as grown d i s l o c a t i o n d e n s i t y i n GaAs (20-25 mm d i a m e t e r ) i s shown i n 150 F i g u r e 4.2 . I t can be seen t h a t Te i s t h e most e f f e c t i v e i n r e d u c i n g the d i s l o c a t i o n d e n s i t y f o l l o w e d by In, Sn and Zn. S e k i 151 and c o w o r k e r s d e t e r m i n e d t h a t Zn, S, Te, A l and N a r e 15 2 15 3 e f f e c t i v e i n r e d u c i n g the d i s l o c a t i o n d e n s i t y . S i l i c o n ' and 154 Boron have a l s o been shown t o have t h e same e f f e c t . D i s l o c a t i o n f r e e GaAs has been o b t a i n e d d o p i n g the m e l t w i t h 15 5 N (20 mm i n d i a m e t e r ) , Te (22 mm i n d i a m e t e r , <111> c r y s t a l ) 1 5 6 , S i (40-50 mm d i a m e t e r ) 1 5 7 , In (50 mm i n ,. . ,35,36,158 d i a m e t e r ) One i m p o r t a n t a s p e c t o f d o p i n g i s t h a t t h e a d d i t i o n p r o d u c e s n- or p - t y p e m a t e r i a l . I f s e m i - i n s u l a t i n g ( S i ) GaAs i s r e q u i r e d , o n l y i s o e l e c t r o n i c i m p u r i t i e s can be added. Indium i s t h e p r o m i s i n g i m p u r i t y t o d e c r e a s e d i s l o c a t i o n d e n s i t y and o b t a i n . . , ^. . ,159-162 s e m i - i n s u l a t i n g m a t e r i a l The p r i m a r y e f f e c t of d o p i n g i s to i n c r e a s e the c r i t i c a l s t r e s s f o r d i s l o c a t i o n g e n e r a t i o n . F o r undoped GaAs the CRSS i s 18 3 7 g/mm w h i l e f o r Te-doped GaAs (2x10 atoms/cm ) t h e CRSS i s 16 3 16 3 40 g/mm . T h i s e f f e c t , as e x p l a i n e d by M i l v i d s k i i e t a l . , a p p e a r s b e c a u s e h i g h t e m p e r a t u r e and low s t r e s s e s c r e a t e 23 f a v o u r a b l e c o n d i t i o n s f o r the f o r m a t i o n of a s t a b l e i m p u r i t y a t m o sphere ( C o t t r e l l a t m o s p h e r e s ) s u r r o u n d i n g moving C [atom- cnr3J F i g u r e 4.2 Mean d e n s i t y o f " g r o w n - i n " d i s l o c a t i o n s i n GaAs s i n g l e c r y s t a l s (20-25 mm i n d i a m e t e r ) grown by t h e C z o c h r a l s k i LEC T e c h n i q u e , as a f u n c t i o n of d o p a n t c o n c e n t r a t i o n : (1) Te, (2) Sn, (3) In, (4) Zn. d i s l o c a t i o n s . T h i s r e d u c e s e f f e c t i v e n e s s o f an e l e m e n t i n 2 g i v e n by a f a c t o r U /D where atoms w i t h d i s l o c a t i o n s and c o e f f i c i e n t . the d i s l o c a t i o n v e l o c i t y . The s t o p p i n g d i s l o c a t i o n movement i s U i s t h e e l a s t i c i n t e r a c t i o n of D i s t h e i m p u r i t y d i f f u s i o n 24 A number of s p e c i f i c mechanisms have been p r o p o s e d t o a c c o u n t f o r t h e e f f e c t of s o l u t e atoms on d i s l o c a t i o n m o b i l i t y . 151 S e k i e t a l . a s s o c i a t e d d i s l o c a t i o n movement w i t h b r e a k i n g o f bonds between s o l u t e e l e m e n t s and t h e m a t r i x , w i t h l i m i t e d 164 s u c c e s s . Sher e t a l . r e l a t e d t h e bond l e n g t h (d) t o d i s l o c a t i o n e n e r g i e s and h a r d n e s s i n s e m i c o n d u c t o r s . They showed t h a t d i s l o c a t i o n e n e r g i e s per u n i t l e n g t h a r e p r o p o r t i o n a l t o -3 -9 -5 -11 d - d and h a r d n e s s t o d - d T h i s a n a l y s i s c o u l d be u s e d t o a c c o u n t f o r the r e d u c t i o n i n d i s l o c a t i o n d e n s i t y i n GaAs 165 1 5 1 1 5 5 by a d d i n g BAs and GaN ' and p r e d i c t s GaP a d d i t i o n s w i l l r e d u c e th e d e n s i t y . The r e d u c t i o n i n d i s l o c a t i o n d e n s i t y by In can be a c c o u n t e d f o r by s o l u t i o n h a r d e n i n g 1 6 6 . In t h i s c a s e t h e s o l u t e e n t i t y i s not t h e In atom a l o n e but c o n s i s t s o f a c l u s t e r o f f i v e atoms. The c l u s t e r i s formed by one In atom bounded by f o u r As atoms, I n A s 4 > i n a t e t r a h e d r a l c o n f i g u r a t i o n w h i c h i s embedded i n a GaAs m a t r i x . These r e s u l t s a r e s u p p o r t e d by the e x p e r i m e n t a l o b s e r v a t i o n s . t h a t the f e e (Ga.In) s u b l a t t i c e p a r a m e t e r i n c r e a s e s l i n e a r l y w i t h In c o n c e n t r a t i o n and t h a t the Ga-As and In-As bond l e n g t h r e m a i n s r o u g h l y c o n s t a n t as x i s v a r i e d i n G a ' 1 X ^ I n X A s 1 6 ^ . The d i l a t i o n of an I n A s 4 t e t r a h e d r a as compared to a GaAs^ t e t r a h e d r a i s 21 % . T h i s i s semi q u a n t i t a t i v e l y c a l c u l a t e d t o have a s t r o n g e r e f f e c t t h a n s o l u t i o n h a r d e n i n g i n m e t a l s . S o l u t e a d d i t i o n s t o the GaAs m e l t can m a r k e d l y i n f l u e n c e t h e b e h a v i o u r of d e v i c e s f a b r i c a t e d on t h e c r y s t a l w a f e r . 25 1) I f s u f f i c i e n t s o l u t e i s added c o n s t i t u t i o n a l s u p e r c o o l i n g c o u l d d e v e l o p a t t h e i n t e r f a c e p r o d u c i n g s o l u t e m i c r o s e g r e g a t i o n i n the c r y s t a l 2) L i n e a g e s t r u c t u r e can be d e v e l o p e d i n t h e c r y s t a l p r o d u c i n g o r i e n t a t i o n v a r i a t i o n s . 3) M i c r o d e f e c t s , i n c l u d i n g p r e c i p i t a t e s can be p r e s e n t i n the c r y s t a l due t o t h e s o l u t e p r e s e n t . As an example, Te a d d i t i o n s at low l e v e l s p r o d u c e s t a c k i n g f a u l t s i n t h e 168-173 c r y s t a l . Te a d d i t i o n s a t h i g h l e v e l s p r o d u c e p r i s m a t i c d i s l o c a t i o n s i n a d d i t i o n to s t a c k i n g 1 7 2 1 7 3 168 f a u l t s ' . S a d d i t i o n s have t h e same e f f e c t H e l i c o i d a l d i s l o c a t i o n s were o b s e r v e d i n GaAs w i t h s m a l l 18 3 a d d i t i o n of Te (2x10 atoms/cm ) and w i t h Sn, S and Zn , . . . . 168,174 a d d i t i o n s C o n t r o l l e d a d d i t i o n s of e l e m e n t s t o GaAs a r e made t o p r o d u c e p- or n - t y p e c r y s t a l s . The e l e m e n t s s e l e c t e d s h o u l d r e s u l t i n the d e s i r e d t y p e of c r y s t a l w i t h m i n i m a l o t h e r d e f e c t s p r o d u c e d , as w e l l as t h e r e d u c t i o n i n d i s l o c a t i o n d e n s i t y . I t has been shown t h a t S i i s t h e b e s t dopant f o r GaAs t o p r o d u c e n - t y p e b e h a v i o u r , 18 3 1 7 5 1 7 6 at l e v e l s o f ne a r 10 atoms/cm , as compared t o Te and S 3 2 In t h i s c a s e d i s l o c a t i o n s d e n s i t i e s o f 3x10 /cm a r e o b t a i n e d w i t h no m i c r o d e f e c t s . At h i g h e r dopant l e v e l s , above 18 3 3x10 atoms/cm , s t r a i g h t and h e l i c o i d a l d i s l o c a t i o n s become p r o n o u n c e d . S i m i l a r r e s u l t s a r e o b t a i n e d w i t h In dopant f o r 19 3 l e v e l s g r e a t e r t h a n 2x10 atoms/cm . At h i g h In l e v e l s 26 20 3 (10 atoms/cm ) t h r e e d i f f e r e n t r e g i m e s of d i s l o c a t i o n s have 17 7 been r e p o r t e d by P i c h a u d e t a l . as shown i n F i g u r e 4.3 . In the f i g u r e t h e In c o n c e n t r a t i o n i s shown t o i n c r e a s e w i t h d i s t a n c e a l o n g t h e growth d i r e c t i o n ( p u l l i n g a x i s ) w h i c h changes the d i s l o c a t i o n d e n s i t y . Position along the pulling axis F i g u r e 4.3 H y p o t h e t i c a l d i s t r i b u t i o n of t h e dopant ( I n ) a l o n g t h e p u l l i n g a x i s of the i n g o t . The i n d i u m c o n c e n t r a - t i o n o s c i l l a t e s a r o und a mean v a l u e w h i c h i s p r o p o r t i o n a l to the d i s t a n c e from t h e top o f the i n g o t . At t h e s e e d end and m i d d l e of t h e c r y s t a l , d i s l o c a t i o n s a r e t a n g l e d w i t h a banded p e r i o d i c a l d i s t r i b u t i o n normal t o the growth d i r e c t i o n . In t h e f i n a l t h i r d o f t h e c r y s t a l t h e bands d i s a p p e a r and t h e d i s l o c a t i o n d e n s i t y i s r e d u c e d . The o s c i l l a t i o n s i n t h e In l e v e l s shown s c h e m a t i c a l l y i n t h e f i g u r e a r e a s s o c i a t e d w i t h c o n v e c t i v e f l o w i n t h e m e l t . 27 S t r i a t i o n o r l i n e a r b o u n d a r i e s a r e o b s e r v e d i n undoped GaAs 17 8 c r y s t a l s . The s t r i a t i o n d e n s i t y can be r e d u c e d by a p p l y i n g a 17 9 m a g n e t i c f i e l d a t k i l o g a u s s l e v e l s t o undoped GaAs , Se 1 7 9 1 8 0 137 doped ' and In doped m a t e r i a l . W i t h a v e r t i c a l m a g n e t i c f i e l d t h e r m a l o s c i l l a t i o n s i n the m e l t due t o c o n v e c t i o n can be r e d u c e d t o l e s s t h a n 0.3°C 1 8 1 but not e l i m i n a t e d 1 8 0 . However, at t h i s time i t i s not c l e a r how the m a g n e t i c f i e l d and c o n v e c t i v e f l o w q u a n t i t a t i v e l y change the c r y s t a l c h a r a c t e r i s t i c . N u m e r i c a l 18 2 183 s i m u l a t i o n s and m o d e l l i n g i n d i c a t e m a g n e t i c f i e l d s a t k i l o g a u s s l e v e l s do not s u p p r e s s c o n v e c t i v e f l o w . R e c e n t r e s u l t s 184 show a 2 KGauss f i e l d can s t a b i l i z e f l o w and t h e r m a l f i e l d s The i n f l u e n c e o f a m a g n e t i c f i e l d on t h e i m p u r i t y d i s t r i b u t i o n i n grown c r y s t a l s i s complex. Carbon c o n c e n t r a t i o n s a r e d e c r e a s e d 181 and Chromium l e v e l s a r e i n c r e a s e d R e s i d u a l i m p u r i t i e s a r e changed, as e v i d e n c e d by a r e d u c t i o n i n e l e c t r i c a l r e s i s t i v i t y of undoped c r y s t a l s 1 8 5 . The a d d i t i o n of In t o GaAs t o r e d u c e d i s l o c a t i o n d e n s i t i e s w i t h o u t c h a n g i n g the e l e c t r i c a l p r o p e r t i e s p r e s e n t s d i f f i c u l t i e s . M a c r o s e g r e g a t i o n of the In a l o n g t h e c r y s t a l a x i s w i l l o c c u r as i n F i g u r e 4.3 . In a d d i t i o n the l a t t i c e s p a c i n g i n GaAs changes w i t h t h e In c o n c e n t r a t i o n w h i c h a f f e c t s t h e q u a l i t y of e p i t a x i a l l a y e r s d e p o s i t e d on the s u r f a c e i n d e v i c e f a b r i c a t i o n . S p e c i f i c a l l y , m i s f i t d i s l o c a t i o n s a r e p r o d u c e d a t t h e SI GaAs s u b s t r a t e and MBE d e p o s i t e d GaAs above a c r i t i c a l d e p o s i t e d l a y e r t h i c k n e s s . The c r i t i c a l l a y e r t h i c k n e s s d e c r e a s e s e x p o n e n t i a l l y w i t h i n c r e a s i n g In c o n c e n t r a t i o n . F o r a s i g n i f i c a n t 28 r e d u c t i o n o f d i s l o c a t i o n s , In c o n c e n t r a t i o n s g r e a t e r t han 19 3 2 0 3 10 atoms/cm a r e r e q u i r e d . At c o n c e n t r a t i o n s o f 10 atoms/cm the c r i t i c a l t h i c k n e s s of d e p o s i t e d m a t e r i a l i s about 1 m i c r o n w h i c h i s s m a l l . The e f f e c t of In a l l o y i n g i n FET c h a r a c t e r i s t i c has been 18 7 examined by H u n t e r e t a l . . U s i n g d i r e c t i o n i m p l a n t a t i o n on In a l l o y e d c r y s t a l w a f e r the FET's showed u n i f o r m p r o p e r t i e s a l o n g t h e w a f e r . Inhomogeneous FET's c h a r a c t e r i s t i c a r e e x p e c t e d from a r r a y s f a b r i c a t e d on w a f e r s w i t h i n h o m o g e n e i t y i n t h e r e s i s t i v i t y . Non u n i f o r m r e s i s t i v i t i e s a l o n g r a d i a l and a x i a l d i r e c t i o n s a r e c h a r a c t e r i s t i c o f In doped GaAs c r y s t a l s w i t h 18 8 d i s l o c a t i o n or d i s l o c a t i o n f r e e m a t e r i a l The u n i f o r m i t y i n the FET p r o p e r t i e s o b t a i n e d by H u n t e r e t a l . c o u l d be a r e s u l t of a p o s t s o l i d i f i c a t i o n a n n e a l i n g p r o c e s s as a r e s u l t o f t h e low growth v e l o c i t i e s and low t h e r m a l g r a d i e n t s d u r i n g c r y s t a l g r o w t h . B u l k a n n e a l i n g has p r o d u c e d u n i f o r m i t y improvements i n 18 8 10 3 undoped GaAs and In doped GaAs 29 CHAPTER 5 MODELS OF DISLOCATION GENERATION IN GaAs M a t h e m a t i c a l models f o r d i s l o c a t i o n g e n e r a t i o n by g l i d e due to t h e r m a l s t r e s s e s have been d e v e l o p e d u s i n g d i f f e r e n t a p p r o a c h e s f o r t h e c a l c u l a t i o n of t h e t h e r m a l f i e l d i n t h e c r y s t a l as w e l l as t h e t h e r m o e l a s t i c s t r e s s f i e l d . 5.1 S t r e s s F i e l d s 1. The t e m p e r a t u r e f i e l d i n t h e g r o w i n g c r y s t a l has been 18 9 c a l c u l a t e d n u m e r i c a l l y a s s u m i n g t h e i n t e r f a c e i s a p a r a b o l o i d o f r e v o l u t i o n . The boundary c o n d i t i o n s were d e t e r m i n e d e x p e r i m e n t a l l y by m e a s u r i n g t h e nea r s u r f a c e t e m p e r a t u r e s at a number of p o s i t i o n s a l o n g the c r y s t a l . The t h e r m o e l a s t i c s t r e s s e s were c a l c u l a t e d a s s u m i n g a c y l i n d r i c a l c r y s t a l w i t h f l a t f a c e s and u s i n g a p l a i n s t r a i n a p p r o x i m a t i o n . The r e s u l t s a r e shown i n F i g u r e 5.1 f o r a <111> grown c r y s t a l . The c a l c u l a t e d i s o t h e r m s and RSS a r e shown as a f u n c t i o n o f p o s i t i o n i n t h e c r y s t a l i n (a) and (b) f o r two c r y s t a l s of d i f f e r e n t d i m e n s i o n s . The number t o the r i g h t of t h e t e m p e r a t u r e s g i v e s t h e CRSS. I t i s a p p a r e n t t h a t t h e r e a r e a few a r e a s where t h e RSS i s l e s s t h a n t h e CRSS (shown by t h e dashed l i n e s ) . The i s o t h e r m s i n p a r t s o f t h e f i g u r e a r e f i r s t c o n c a v e f o l l o w i n g t h e i n t e r f a c e shape and change t o convex a f t e r t h e 1200°C i s o t h e r m . Growth c o n d i t i o n s and c r y s t a l c h a r a c t e r i s t i c s a r e not g i v e n . In p a r t s ( c ) and (d) o f the f i g u r e 30 0-V 0.8 rIK (I 0.1/ fi• g 9, 710 F i g . 5.1 (a) and (b) - I s o t h e r m s and s h e a r s t r e s s t o p o g r a p h y i n two g a l l i u m a r s e n i d e s i n g l e c r y s t a l grown under d i f f e r e n t c o n d i t i o n s . The f i g u r e s n e x t t o the c u r v e s a r e x v a l u e s (kg/mm ) c o r r e s p o n d i n g t o t h e c o n s t a n t - s t r e s s l i n e s . The dashed c u r v e s o u t l i n e r e g i o n s i n w h i c h th e e f f e c t i v e s t r e s s e s a r e l o w e r t h a n t h e r e d u c e d y i e l d s t r e s s . The f i g u r e s i n p a r e n t h e s e s a l o n g t h e o r d i n a t e , a x i s g r e t h e e x p e r i m e n t a l c r i t i c a l s t r e s s e s i n 10 kg/mm . ( c ) and (d) - D i s t r i b u t i o n of s h e a r s t r e s s e s T., and T 3 ( c ) and d i s l o c a t i o n d e n s i t y (d) o v e r t h e c r o s s - s e c t i o n of a g a l l i u m a r s e n i d e s i n g l e c r y s t a l . 31 the mean RSS shows the t y p i c a l W- shaped d i s t r i b u t i o n of the d i s l o c a t i o n s a l o n g t h e w a f e r d i a m e t e r ; t h e s t r e s s symmetry i s not o b t a i n e d . The e f f e c t of growth c o n d i t i o n s i s n o t a n a l y s e d . 2. The t e m p e r a t u r e f i e l d i n g r o w i n g GaAs has been d e t e r m i n e d a n a l y t i c a l l y by J o r d a n et a 1 . 1 3 7 • 1 4 8 • 1 9 0 . They u s e d a q u a s i - s t e a d y s t a t e a p p r o x i m a t i o n of the h e a t c o n d u c t i o n e q u a t i o n i n c y l i n d r i c a l c o o r d i n a t e s . F o r t h e boundary c o n d i t i o n s t h e y assumed a c o n s t a n t t e m p e r a t u r e a t a f l a t s o l i d / l i q u i d i n t e r f a c e and c o n v e c t i o n / r a d i a t i o n heat l o s s at t h e s u r f a c e o f t h e c r y s t a l f o l l o w i n g Newton's Law of c o o l i n g . The h e a t t r a n s f e r c o e f f i c i e n t s u sed were e v a l u a t e d s e p a r a t e l y as a f u n c t i o n of temper- 13 9 191 a t u r e ' . U s i n g t h e c a l c u l a t e d t h e r m a l f i e l d s , t h e s t r e s s e s i n the c r y s t a l were c a l c u l a t e d u s i n g a p l a i n s t r a i n a p p r o x i m a t i o n , and t h e RSS was d e t e r m i n e d f o r a <100> o r i e n t e d c r y s t a l . The l o c a l d i s l o c a t i o n d e n s i t y was assumed to be p r o p o r t i o n a l t o t h e t o t a l RSS. T h i s was d e f i n e d as t h e sum of the a b s o l u t e v a l u e s o f RSS a c t i n g on {111} <110> s l i p s y s t e m . The r e s u l t s a r e shown i n F i g u r e 5.2 i n which t h e t o t a l RSS (TRSS) i s p l o t t e d on a t r a n s v e r s e p l a n e of a c y l i n d r i c a l c r y s t a l . T y p i c a l f o u r - f o l d symmetry i s o b s e r v e d w i t h s t r e s s minima midway between the o u t s i d e and c e n t r e of the c r y s t a l a l o n g t h e <110> d i r e c t i o n s . The p r i n c i p a l c o n c l u s i o n s d e r i v e d from t h e model f o l l o w : 1) D o u b l i n g t h e r a d i u s of the c y l i n d r i c a l c r y s t a l from 2 to 4 cm more t h a n d o u b l e s the TRSS. 2) The d i s l o c a t i o n d e n s i t y i n c r e a s e s w i t h i n c r e a s i n g h e a t t r a n s f e r c o e f f i c i e n t s . 32 3) The d i s l o c a t i o n d e n s i t y can h e a t t r a n s f e r c o e f f i c i e n t i s one o r d e r o f magnitude s m a l l t h e c a l c u l a t i o n s . be r e d u c e d t o z e r o i f t h e s m a l l enough, e s t i m a t e d as er t h a n t h e v a l u e s used i n 3. The t e m p e r a t u r e f i e l d i s c a l c u l a t e d n u m e r i c a l l y and t h e t h e r m a l s t r e s s e s d e t e r m i n e d . However, i n t h i s c a s e s t r e s s r e l a x a t i o n d u r i n g growth i s i n c l u d e d i n t h e a n a l y s i s (Vakhrameev 192 e t a l . ). The t h e r m a l f i e l d i s d e t e r m i n e d by s o l v i n g t h e s t e a d y s t a t e t h e r m a l c o n d u c t i o n e q u a t i o n f o r t h e e n t i r e g r o w t h s y s t e m i n c l u d i n g c r y s t a l , m e l t , gas and c r u c i b l e . The e n c a p s u l a n t B O 19 3 i s n e g l e c t e d The t h e r m o e l a s t i c s t r e s s f i e l d i s o b t a i n e d by s o l v i n g the d i s p l a c e m e n t t h e r m o e l a s t i c e q u a t i o n by a f i n i t e 19 4 d i f f e r e n c e method R e s u l t s a r e shown i n F i g u r e 5 . 3 ( a - c ) . In F i g u r e 5.3(a) and (b) the d i s l o c a t i o n d e n s i t y i s shown as a f u n c t i o n o f t e m p e r a t u r e f o r a B u r g e r v e c t o r p e r p e n d i c u l a r t o and i n c l i n e d to the growth d i r e c t i o n <111>. A c o n s i d e r a b l e f r a c t i o n of t h e d i s l o c a t i o n s i s o b s e r v e d t o be on t h e p e r i p h e r y o f t h e c r y s t a l a t t e m p e r a t u r e s w e l l below the m e l t i n g t e m p e r a t u r e . F i g u r e 5 . 3 ( c ) shows d i s l o c a t i o n d e n s i t i e s , t h e r m o e l a s t i c s t r e s s e s , r e l a x e d s t r e s s e s and CRSS f o r undoped GaAs a l o n g the a x i s o f t h e c r y s t a l . Two w e l l d e f i n e d r e g i o n s a r e o b s e r v e d where d i s l o c a t i o n s a r e formed, one n e a r th e c r y s t a l l i z a t i o n f r o n t at 1238°C and t h e o t h e r n e a r 1140°C. T h e r m o e l a s t i c s t r e s s e s i n t h i s c a s e a r e r e l a x e d about 80 % . The g r o w t h c o n d i t i o n s a r e not g i v e n and t h e e f f e c t o f g r o w t h p a r a m e t e r s i s not d i s c u s s e d . 33 F i g u r e 5.2 T R S S c o n t o u r s f o r the t o p w a f e r o f a <001> GaAs b o u l e . 34 a b c 5.3 D i s t r i b u t i o n of t h e c a l c u l a t e d d i s l o c a t i o n d e n s i t y i n a g a l l i u m a r s e n i d e s i n g l e c r y s t a l b e i n g grown i n t h e <111> d i r e c t i o n f o r s l i p s y s t e m s w i t h B u r g e r s d i s l o c a t i o n v e c t o r s p e r p e n d i c u l a r (a) and i n c l i n e d (b) t o t h e growth a x i s at t h e f o l l o w i n g t e m p e r a t u r e s : 1) 1234, 2) 1200, 3) 1140, 4) 1050 C - T. ( c ) D i s t r i b u t i o n o f c a l c u l a t e d d i s l o c a t i o n d e n s i t y 1) t h e r m o e l a s t i c s t r e s s , 2) r e l a x e d t h e r m a l s t r e s s , 4) and c r i t i c a l s t r e s s f o r the f o r m a t i o n of d i s l o c a t i o n i n undoped GaAs a t t h e c o r r e s p o n d - i n g t e m p e r a t u r e , 3) a l o n g th e a x i s o f a s i n g l e c r y s t a l d u r i n g i t s g r o w t h ; CF) c r y s t a l l i z a t i o n f r o n t . 35 195 4. Dussaux has employed a f i n i t e e l e m e n t method t o o b t a i n b o t h t h e r m a l and s t r e s s f i e l d s . The t h e r m a l f i e l d was c a l c u l a t e d by s o l v i n g the s t e a d y h e a t c o n d u c t i o n e q u a t i o n i n t h e c r y s t a l s u b j e c t to Newton's Law o f C o o l i n g a t the c r y s t a l s u r f a c e as done . . , . . 137,148,190 by J o r d a n e t a l . The n u m e r i c a l scheme c h o s e n p e r m i t s more r e a l i s t i c b o u n d a r y c o n d i t i o n s t o be used s u c h as t e m p e r a t u r e g r a d i e n t s o u t s i d e t h e c r y s t a l and t e m p e r a t u r e d e p e n d e n t h e a t t r a n s f e r c o e f f i c i e n t s . In a d d i t i o n d i f f e r e n t g e o m e t r i e s can be c o n s i d e r e d . The s t r e s s f i e l d i s c a l c u l a t e d u s i n g the minimum e n e r g y p r i n c i p l e method and an a x i s y m m e t r i c f i e l d i s assumed. From the s t r e s s components the Von N i s e s s t r e s s i s d e r i v e d . S t r e s s e s a r e not r e s o l v e d i n the s l i p s y s t e m . The Von M i s e s s t r e s s c o n t o u r s i n t h e c r y s t a l f o r a f i x e d b o r o n o x i d e l a y e r , and t h e r m a l g r a d i e n t s a r e p r e s e n t e d f o r two c r y s t a l d i a m e t e r s and d i f f e r e n t c r y s t a l l e n g t h s . The model i s employed to compare the LEC and LEK ( l i q u i d e n c a p s u l a t e d K r y o p o u l o s ) methods and not t o an e x t e n s i v e a n a l y s i s o f d i s l o c a t i o n g e n e r a t i o n i n LEC g r o w t h . 5. T h e r m a l s t r e s s e s have been c a l c u l a t e d u s i n g a n a l y t i c a l a p p r o a c h e s i n which th e p l a i n s t r a i n a p p r o x i m a t i o n i s not assumed. I n s t e a d of p l a i n s t r a i n an a x i s y m m e t r i c a p p r o x i m a t i o n i s u s e d . Q u a l i t a t i v e good agreement w i t h p l a i n s t r a i n and FEM 19 6 c a l c u l a t i o n s i s o b t a i n e d . U s i n g t h e same a x i s y m m e t r i c 19 7 198 a p p r o x i m a t i o n ' , i t was shown t h a t t h e p l a i n s o l u t i o n s a r e not a l w a y s v a l i d i n d e t e r m i n i n g t h e t h e r m a l s t r e e s s e s i n t h e 36 c r y s t a l . T hese c o n c l u s i o n s a r e o b t a i n e d f o r s e m i - i n f i n i t e g e o m e t r i e s . 199 6. G a l a k t i o n o v and T r o p p have c o n s i d e r e d a non s t a n d a r d a p p r o a c h t o i n t r o d u c e the e f f e c t o f a x i a l g r a d i e n t s . They assume t h a t t h e a x i a l g r a d i e n t o f t h e t o t a l s t r e s s e s ( t r a n s i e n t and r e s i d u a l ) a r e n u m e r i c a l l y e q u a l t o the t h e r m o e l a s t i c s t r e s s e s p r o d u c e d by a h y p o t h e t i c a l t e m p e r a t u r e f i e l d g i v e n by t h e a x i a l g r a d i e n t o f t h e r e a l f i e l d . In e q u a t i o n form t h i s i s e x p r e s s e d as 3 9 0. . [ T ( x , y , z ) ] = O t T ( x , y , z ) ] 3z 1 J 1 J 3z The method i s us e d t o c a l c u l a t e t h e r e s i d u a l s t r e s s e s i n i n f i n i t e c y l i n d e r s . 5 . 2 C a l c u l a t i o n of D i s l o c a t i o n D e n s i t i e s The d i s l o c a t i o n d e n s i t y a t a p o i n t i n a g r o w i n g c r y s t a l i s a f u n c t i o n o f t h e l o c a l s t r e s s e s g e n e r a t e d by t h e t h e r m a l s t r a i n s . The d e n s i t y may be assumed to be p r o p o r t i o n a l t o t h e maximum RSS (MRSS) or t o a l e s s e r v a l u e i f t h e s t r e s s i s c o n s i d e r e d t o be r e l i e v e d by p l a s t i c f l o w . T h i s has been c o n s i d e r e d i n a number o f d i f f e r e n t ways : 1. J o r d a n e t a l . (model 2) assumed t h a t t h e t h e r m o e l a s t i c s t r e s s e s were p a r t i a l l y r e l i e v e d by p l a s t i c f l o w but made no e s t i m a t e of t h e amount r e l i e v e d . The f l o w was assumed to be p r o p o r t i o n a l to t h e TRSS. 37 2. Vakhrameev e t a l . (model 3) c a l c u l a t e d t h a t 80 * o f t h e t h e r m o e l a s t i c s t r e s s Is r e l a x e d by p l a s t i c f l o w . 3. B i l l i g i n 1956 and Indenbom i n 1957 p r o p o s e d t h a t s t r e s s e s a r e c o m p l e t e l y r e l a x e d . 12 9 B i l l i g e s t i m a t e d t h e d i s l o c a t i o n d e n s i t y u s i n g t h e f o l l o w i n g s i m p l e a n a l y s i s . C o n s i d e r a t h i n s l a b o f t h i c k n e s s 6r and l e n g t h z. The a p p l i c a t i o n of a t e m p e r a t u r e g r a d i e n t 6 T / 6 r w i l l expand the h e a t e d s u r f a c e by 6r = az 6T. T h i s e x p a n s i o n w i l l bend t h e s l a b g i v i n g a c u r v a t u r e r a d i u s 1/R = ( 1 / z ) x ( 6 z / 6 r ) . I f s t r e s s e s a r e r e l a x e d by p l a s t i c d e f o r m a t i o n a number of edge d i s l o c a t i o n s a r e f o r m e d . The d e n s i t y n i s g i v e n by t h e c u r v a t u r e and B u r g e r ' s v e c t o r , n = 1/Rb, t h e r e f o r e n = (a/b) (6T/6r) U s i n g t h i s r e l a t i o n B i l l i g e s t i m a t e d d i s l o c a t i o n d e n s i t i e s 4 2 i n Ge o f the o r d e r of 3.2x10 /cm f o r g i v e n g r o w t h c o n d i t i o n s . Indenbom p r o p o s e d a s i m i l a r e x p r e s s i o n i n w h i c h the a x i a l t e m p e r a t u r e g r a d i e n t i s c o n s i d e r e d . T h i s i s g i v e n by n = (1/b) [ d ( a T ) / d t ] The above e x p r e s s i o n i s p a r t o f a more complex e q u a t i o n t h a t i n v o l v e s a t e n s o r i a l d i s l o c a t i o n d e n s i t y n, g i v e n by n = - g r a d T X a 38 where a i s t h e t e n s o r of the c o e f f i c i e n t s of l i n e a r t h e r m a l 2 0 2 2 0 3 e x p a n s i o n ' The c o n t r i b u t i o n o f t h e r a d i a l and a x i a l t e m p e r a t u r e g r a d i e n t s t o t h e t o t a l d i s l o c a t i o n d e n s i t y i s not , 119,204 c l e a r Indenbom a l s o p r o p o s e d a n o t h e r e x p r e s s i o n t o e s t i m a t e t h e t o t a l d i s l o c a t i o n d e n s i t y from p l a s t i c d e f o r m a t i o n e n = e / b L where L i s t h e d i s l o c a t i o n l e n g t h w h i c h , f o r Ge and S i , i s o f t h e 10 6 2 0V o r d e r o f 0.25 mm. The Indenbom e q u a t i o n s were e x t e n t e d ' by i n t r o d u c i n g t h e CRSS i n the g l i d e p l a n e t o o b t a i n t h e f o l l o w i n g e x p r e s s i o n f o r t h e d i s l o c a t i o n d e n s i t y •+ _ n = ( a / b ) V T n - 2 ( T / G b ) ( l / D ) c r w h i c h r e d u c e s t o t h e e q u a t i o n g i v e n by Indenbom when the CRSS i s -y z e r o . In t h e e q u a t i o n VTn i s t h e mean a x i a l t e m p e r a t u r e g r a d i e n t , D i s t h e c r y s t a l d i a m e t e r and G i s t h e s h e a r modulus. The d i s l o c a t i o n d e n s i t i e s c a l c u l a t e d w i t h t h i s e q u a t i o n a r e compared t o measurements on GaAs i n T a b l e 5.1. The columns headed N l and N2 a r e c a l c u l a t e d v a l u e s f o r t h e f i r s t and s e c o n d terms o f the above e q u a t i o n . T h i s e q u a t i o n i s n o t used i n t h e p r e s e n t i n v e s t i g a t i o n b e c a u s e i t i s i n c o n s i s t e n t w i t h the e v a l u a t i o n of d i s l o c a t i o n d e n s i t i e s from s t r e s s e s i n t h e c r y s t a l . T h i s i s b e c a u s e a c o n s t a n t a x i a l t h e r m a l g r a d i e n t does not p r o d u c e s t r e s s e s . 39 In summary from t h e r e p o r t e d r e s u l t s I t i s not c l e a r t o what e x t e n t t h e t h e r m o e l a s t i c s t r e s s e s a r e r e l i e v e d . In d e r i v i n g d i s l o c a t i o n d e n s i t i e s a d j u s t a b l e p a r a m e t e r s a r e i n t r o d u c e d i n p s e u d o - e m p i r i c a l e q u a t i o n s w h i c h a r e n o t t e s t e d under c o n t r o l l e d l a b o r a t o r y c o n d i t i o n s . In a d d i t i o n , i n the e q u a t i o n s i n w h i c h d i s l o c a t i o n d e n s i t i e s a r e d i r e c t l y p r o p o r t i o n a l t o t h e r m a l g r a d i e n t s , the r o l e of s t r e s s e s i n g e n e r a t i n g d i s l o c a t i o n s i s not c l e a r . T a b l e 5.1 O b s e r v e d (N ) and C a l c u l a t e d (N, and Nn) D i s l o c a t i o n exD 1 * 2 D e n s i t i e s f o r G a l l i u m A r s e n i d e S i n g l e C r y s t a l s . VT N ' 3 6 X 0 2 |D,cm | | o i l N i . C B > I N 9,cm | Growth method | deg/cm cm 1,0 142 5 1 0 3 1,7 1 0 4 2,2 10* 2,4 140 "3-5 10* 2,0 10* 2,2 10* 2,4 120 ~ l - 2 10* 1,7 10* 1,9 10* The C z o c h r a l s k i 2,4 120 "0,9 10* 1,7 10* 1,9 10* method 2,4 50 "5-6 1 0 3 5,7 10^ 7,8 10^ 2,7 190 5 10 2,7 10 2,9 10 0,8 450 "1 1 0 5 0,6 1 0 5 0,7 10^ 0,8 380 7-9 10* 5,3 10* 5,9 1 0 4 F r e e z o n a l 0,8 290 4-6 10 3,9 10 4,5 10 m e l t i n g 0,8 210 2-3 10 2,6 10 3,2 10 Note and N g were c a l c u l a t e d w i t h E q . ( 2 ) and Indgnbomjs e q u a t i o n , r e s p e c t i v e l y ; b = 4,10 cm |21] , a =6,2 10 deg and i t was assumed t h a t T /G - 1 10 . er 40 CHAPTER 6 OBJECTIVES In t h e p r e v i o u s c h a p t e r s i t has been p o i n t e d out t h a t d i s l o c a t i o n d e n s i t i e s a t t h e l e v e l s u s u a l l y p r e s e n t i n LEC GaAs a r e d e t r i m e n t a l to t h e p r e f o r m a n c e o f f a b r i c a t e d d e v i c e s . New t e c h n o l o g i c a l a d v a n c e s i n t h e GaAs m i c r o and o p t o e l e c t r o n i c i n d u s t r y r e q u i r e GaAs w a f e r s o f low d i s l o c a t i o n d e n s i t y or f r e e o f d i s l o c a t i o n s . D i s l o c a t i o n s i n LEC GaAs a r e g e n e r a t e d d u r i n g growth m a i n l y due t o t h e r m a l s t r e s s e s . On the b a s i s of a g l i d e mechanism t h e r m o e l a s t i c models have been d e v e l o p e d t o d e s c r i b e t h e d i s l o c a t i o n d i s t r i b u t i o n i n as grown c r y s t a l s . These models employ a n a l y t i c a l and n u m e r i c a l methods. The models can q u a l i t a t i v e l y d e r i v e t h e W-shape and f o u r - f o l d symmetry o f the d i s l o c a t i o n d i s t r i b u t i o n . However t h e y c a n n o t d e t e r m i n e t h e a x i a l d i s t r i b u t i o n i n t h e g r o w i n g c r y s t a l , nor can t h e y be u s e d t o d e v e l o p changes i n growth c o n d i t i o n s t o r e d u c e or e l i m i n a t e d i s l o c a t i o n s i n GaAs. T h i s r e s u l t s from t h e a n a l y t i c a l s o l u t i o n l i m i t a t i o n s , b o t h p l a i n s t r a i n and a x i s y m m e t r i c a p p r o x i m a t i o n s , and t h e l i m i t e d s c o p e of the Dusseaux n u m e r i c a l model. The models do not a d e q u a t e l y t a k e i n t o a c c o u n t t h e p r i m a r y g r o w t h p a r a m e t e r s d u r i n g c r y s t a l g r o w t h p r o d u c i n g t h e t h e r m a l f i e l d s on w h i c h t h e models a r e b a s e d . E x p e r i m e n t a l measurements 41 o f t h e t h e r m a l f i e l d s , as a f u n c t i o n o f the growth p a r a m e t e r s , to i n c o r p o r a t e i n t o t h e model a r e v e r y d i f f i c u l t t o make i n h i g h p r e s s u r e g r o w e r s . The main o b j e c t i v e of t h i s i n v e s t i g a t i o n i s t o f u r t h e r d e v e l o p t h e m a t h e m a t i c a l model f o r d i s l o c a t i o n g e n e r a t i o n and m u l t i p l i c a t i o n t a k i n g i n t o a c c o u n t a l l o f t h e v a r i a b l e s r e l a t e d to c r y s t a l g r o w t h . E x p e r i m e n t a l o b s e r v a t i o n s r e p o r t e d i n t h e l i t e r a t u r e of t h e r m a l f i e l d s and p h y s i c a l q u a n t i t i e s o f t h e GaAs and B O , and made i n a h i g h p r e s s u r e grower w i l l be u s e d t o e s t a b l i s h the b o u n d a r y c o n d i t i o n s i n t h e model. The model w i l l be a b l e t o d e t e r m i n e the a x i a l d i s l o c a t i o n d i s t r i b u t i o n and w i l l have t h e f l e x i b i l i t y t o i n c o r p o r a t e any change i n t h e g r o w t h c o n d i t i o n s . The model w i l l be used t o a s s e s s t h e d i s l o c a t i o n d e n s i t y i n c r y s t a l s w i t h t h e i n t r o d u c t i o n o f m o d i f i c a t i o n s o f the t h e r m a l f i e l d s i n t h e c r y s t a l g r o w e r s . The o b j e c t i v e would be t o d e v e l o p p r o c e d u r e s f o r l o w e r i n g or e l i m i n a t i n g d i s l o c a t i o n d e n s i t i e s i n th e c r y s t a l w i t h o u t i n t r o d u c i n g o t h e r d e f e c t s w h i c h a f f e c t t h e e l e c t r i c a l b e h a v i o u r of t h e c r y s t a l s . The i n v e s t i g a t i o n was c a r r i e d out i n t h e f o l l o w i n g s e q u e n c e : 1. Based on a g l i d e mechanism due t o t h e r m a l s t r e s s e s , m a t h e m a t i c a l models f o r d i s l o c a t i o n g e n e r a t i o n i n LEC GaAs d u r i n g growth and s u b s e q u e n t c o o l i n g , were f o r m u l a t e d , d e v e l o p e d and v a l i d a t e d . 42 Using the models, most important f a c t o r s a f f e c t i n g d i s l o c a t i o n g e n e r a t i o n i n GaAs c r y s t a l s d u r i n g and a f t e r growth, were determined. Based on the above r e s u l t s and the thermal h i s t o r y the r o l e of thermal s t r e s s e s i n the g e n e r a t i o n of d i s l o c a t i o n s i n LEC c r y s t a l s , were c l a r i f i e d . 43 CHAPTER 7 FORMULATION OF THE MODELS The p r o d u c t i o n o f c r y s t a l s can be d i v i d e d i n t o two p a r t s c r y s t a l g rowth, w h i c h i n c l u d e s t h e s o l i d i f i c a t i o n p r o c e s s , and t h e s u b s e q u e n t c r y s t a l c o o l i n g t o ambient t e m p e r a t u r e . B o t h p a r t s of t h e p r o c e s s a r e m o d e l l e d s e p a r a t e l y . In t h e two models th e b a s i c a s s u m p t i o n i s made t h a t t h e t e m p e r a t u r e f i e l d s and s t r e s s f i e l d s a r e d e c o u p l e d 2 ^ ® . A s i m p l i f i e d f l o w c h a r t o f t h e models w h i c h d e r i v e t h e s t r e s s f i e l d s from t h e t h e r m a l f i e l d s i s shown i n F i g u r e 7.1 The f l o w c h a r t f o r b o t h p a r t s o f t h e p r o c e s s a r e i d e n t i c a l ; t h e d i f f e r e n c e between b o t h models i s the way the t h e r m a l f i e l d s a r e c a l c u l a t e d . At a s p e c i f i c t i m e d u r i n g growth, t h e s t r e s s f i e l d s a r e d e r i v e d from the c a l c u l a t e d t h e r m a l f i e l d s . The s t r e s s components a r e t h e n u s e d t o c a l c u l a t e t h e Von M i s e s S t r e s s or t h e R e s o l v e d Shear S t r e s s (RSS) a t a g i v e n p o i n t i n t h e c r y s t a l . The RSS a r e t h e p r o j e c t i o n s of t h e s t r e s s t e n s o r i n s p e c i f i c p l a n e s and d i r e c t i o n s . In our c a s e t h o s e p l a n e s and d i r e c t i o n s a r e g i v e n by t h e g l i d e s y s t e m of GaAs {111} <110>. F o r a c r y s t a l grown i n t h e [001] d i r e c t i o n , from t h e t w e l v e c o m b i n a t i o n s o f p l a n e s and d i r e c t i o n s t h e r e a r e t e n d i f f e r e n t RSS components. I t i s assumed i n t h e model t h a t when the maximum component o f RSS i s g r e a t e r t h a n a c r i t i c a l v a l u e , d i s l o c a t i o n s a r e g e n e r a t e d and m u l t i p l y d u r i n g g l i d e . The p a r t i c u l a r p l a n e s and d i r e c t i o n s i n w h i c h t h e 44 T E M P E R A T U R E F I E L D STRESS F I E L D V O N MISES STRESS G L I D E M O D E R E S O L V E STRE ;D S H E A R ;ss M A X I M U M S H E A R S' R E S O L V E D rRESS N O DISLOCATIONS F i g u r e 7.1 Flow chart of the model used to d e r i v e the s t r e s s f i e l d s . 45 RSS a r e maxima w i l l d e t e r m i n e the g l i d e mode and c r y s t a l l o g r a p h y o f t h e d i s l o c a t i o n s b e i n g g e n e r a t e d . I t i s a l s o assumed t h a t when one mode o p e r a t e s , the s t r e s s i s r e l a x e d and t h e r e i s no f u r t h e r g l i d e i n o t h e r d i r e c t i o n s . 7.1 Model o f c r y s t a l growth 7.1.1 T e m p e r a t u r e F i e l d 7.1.1.1 G o v e r n i n g E q u a t i o n s A c o m p l e t e a n a l y s i s of t h e t e m p e r a t u r e f i e l d i n a LEC-GaAs c r y s t a l d u r i n g growth r e q u i r e s t h e s o l u t i o n of a S t e f a n ' s p r o b l e m w i t h complex b o u n d a r y c o n d i t i o n s . The e x a c t s o l u t i o n of t h i s p r o b l e m i s time dependent and i n c l u d e s the f o u r p h a s e s , c r y s t a l , gas, e n c a p s u l a n t and m e l t as w e l l as t h e mold and the h e a t i n p u t / o u t p u t to the mold. The h e a t t r a n s f e r mechanisms i n c l u d e c o n d u c t i o n , r a d i a t i o n and c o n v e c t i o n . To t r e a t t h i s p r o b l e m m a t h e m a t i c a l l y the f o l l o w i n g a s s u m p t i o n s a r e made : 1. The t e m p e r a t u r e f i e l d i n t h e c r y s t a l i s a x i a l l y s y m m e t r i c , i . e . T = T ( r , z , t ) 2. The i n t e r f a c e shape i s u n a l t e r e d by d i s s i p a t i o n of l a t e n t h e a t and r e m a i n s at the m e l t i n g p o i n t , T M P 3. Heat t r a n s f e r a t the i n t e r f a c e o f the c r y s t a l ( e x c l u d i n g t h e s o l i d i f y i n g i n t e r f a c e ) f o l l o w s Newton's Law o f C o o l i n g 46 4. The heat t r a n s f e r c o e f f i c i e n t and a m bient t e m p e r a t u r e a r e a l l o w e d t o v a r y a l o n g t h e l a t e r a l s u r f a c e of t h e c r y s t a l , i . e . i n c o n t a c t w i t h t h e e n c a p s u l a n t o r t h e g a s e o u s atmosphere 5. The t e m p e r a t u r e f i e l d i s changed by a change i n c r y s t a l l e n g t h w i t h t i m e , i . e . a q u a s i - s t e a d y s t a t e a p p r o x i m - a t i o n . W i t h the above a s s u m p t i o n s th e non- d i m e n s i o n a l e q u a t i o n o f h e a t c o n d u c t i o n , w i t h r e s p e c t t o a c o o r d i n a t e s y s t e m f i x e d t o the s o l i d i f y i n g i n t e r f a c e , as shown i n F i g u r e 7.2 , t a k e s t h e form : 3 23 l 36 3 26 36 L (6) = -r- + + r - - 2V = o (a) 3p p 3p 3? 3£ (7.1) T - T MP 0 p = r / r Q , £ = z / r Q (b) where 2V = v r Q / K , T Q i s t h e r e f e r e n c e t e m p e r a t u r e , r Q i s t h e c r y s t a l r a d i u s , v the growth v e l o c i t y and K the t h e r m a l c o n d u c t i v i t y . S o l u t i o n s must a l s o s a t i s f y t h e f o l l o w i n g g e n e r a l b o u n d a r y c o n d i t i o n s : i . D i r i c h l e t c o n d i t i o n At t h e s o l i d - l i q u i d i n t e r f a c e g i v e n by t h e s u f a c e o f r e v o l u t i o n f ( r , z ) = 0 , r < r ^ 6 = 1 (7.2) lz=0 r melt Figure 7.2 C r y s t a l c o n f i g u r a t i o n and c o o r d i n a t e the mathematical model. system used i n 48 i i . Neumann c o n d i t i o n At any p o i n t S on t h e c y l i n d r i c a l s u r f a c e o f t h e c r y s t a l , 33 33 ( — n D + — n F ) _ n = - h ( z ) r . r f i - 3 n U ) l (7.3) 9 p P 3 ? £, S 0 S a where 3 ( z ) i s t h e n o r m a l i z e d t e m p e r a t u r e o f t h e ambient a d j a c e n t t o t h e c r y s t a l s u r f a c e at a d i s t a n c e z from t h e i n t e r f a c e and 3 g i s t h e n o r m a l i z e d t e m p e r a t u r e of t h e c r y s t a l a t t h a t p o i n t ; h ( z ) i s t h e h e a t t r a n s f e r c o e f f i c i e n t between c r y s t a l s u r f a c e and ambient at t h e same p o i n t . The n u m e r i c a l scheme employed t o s o l v e t h i s i s g i v e n below. 7.1.1.2 F i n i t e E l e m e n t E q u a t i o n s Among t h e d i f f e r e n t a p p r o a c h e s f o r o b t a i n i n g t h e e l e m e n t e q u a t i o n s i n a f i n i t e e l e m e n t method, t h e G a l e r k i n ' s method was s e l e c t e d . In t h i s method, a f t e r t h e whole domain i n w h i c h a s o l u t i o n f o r t h e f i e l d v a r i a b l e 3 i s r e q u i r e d i s d i s c r e t i z e d i n s m a l l e l e m e n t s , t h e ele m e n t e q u a t i o n i s o b t a i n e d by p e r f o r m i n g the f o l l o w i n g i n t e g r a t i o n o v e r each e l e m e n t , S. . M 3 ( e ) ) N < e ) d D ( e ) D ( e ) i 0 w i t h (e) [N] {3} (e) [ H 2 N 0 . . N ] < g ^ g (7.4) 49 where t h e N's a r e t h e i n t e r p o l a t i o n f u n c t i o n s , t h e 3 ^ a r e t h e n o d a l f i e l d v a l u e s i n element (e) and g i s the d e g r e e o f freedom of t h e e l e m e n t . The i n t e r p o l a t i o n f u n c t i o n s must be s u c h t h a t t h e r e i s c o n t i n u i t y o f the f i e l d v a r i a b l e and o f i t s p a r t i a l d e r i v a t i v e s up t o one o r d e r l e s s t h a n the h i g h e s t o r d e r d e r i v a t i v e i n t h e o p e r a t o r . T h i s means t h a t i n t e r p o l a t i o n f u n c t i o n s f o r a C* p r o b l e m a r e a t f i r s t a p p a r e n t l y r e q u i r e d . However i n t e g r a t i o n by p a r t s and t h e use of the b o u n d a r y c o n d i t i o n s t r a n s f o r m t h e p r o b l e m i n s u c h a way as to show t h a t l e s s s t r i c t i n t e r p o l a t i o n f u n c t i o n s can be u s e d . These c a l c u l a t i o n s a r e p r e s e n t e d i n A p p e n d i x II where i t i s shown t h a t t h e i n i t i a l p r o b l e m g i v e n by E q u a t i o n (7.4) i s t r a n s f o r m e d i n t o t h e f o l l o w i n g e q u i v a l e n t e q u a t i o n 3N 3 g ( e ) 3N. 3 3 ( 6 ) 3 g ( e ) ( + - + 2V N. ) dV(e) + D(e) 3P 9P 3£ 9? 1 35 . + r h ( z ) f N. 3 ( E ) d S ( e ) = s ( e ) = r Q h ( z ) B ( z ) f N . d S ( 6 ) (7.5) J s ( e ) where S^ e' i s t h e s u r f a c e a r e a of e l e m e n t (e) and dS i s t h e e l e m e n t a l s u r f a c e . I t i s now c l e a r from t h e above e q u a t i o n t h a t i n t e r p o l a t i o n f u n c t i o n s f o r C^ 0^ p r o b l e m s a r e r e q u i r e d . 5 0 A s u i t a b l e element shape chosen here f o r s u b d i v i s i o n of axisymmetric bodies i s the axisymmetric r i n g element with t r i a n g u l a r c r o s s - s e c t i o n . The f i e l d v a r i a b l e i s assumed to vary l i n e a r l y i n s i d e the element and a three node element i s chosen. For these l i n e a r t r i a n g u l a r t o r o i d a l elements, i n t e r p o l a t i o n f u n c t i o n s f o r C^°^ problems are the n a t u r a l or area coodinates f u n c t i o n . The L. 's are d e f i n e d as L. ( P, C ) 2 A ( a i + bi p + c ) (7.6) where a . i = PjSc " P k ? j : c i " P k - p j a n d b i = ? j " ? k ( 7 - 7 ) where ( i , j , k) have module 3 and permute c y c l i c a l l y , ( Pj , C ^ are the c o o r d i n a t e s of the nodal p o i n t s ( v e r t i c e s ) of a t r i a n g l e numbered counter-clockwise 2 A ( 7 . 8 ) A i s the area of the t r i a n g l e The temperature w i t h i n t h i s element i s expressed by the equation 3 (e) (e) [ L 1 L2 L 3 ] <! 02 (e) ( 7 . 9 ) 51 where the 3 ^ ' s a r e the n o d a l v a l u e s of t h e t e m p e r a t u r e In e l e m e n t (e) i n t h e l o c a l n u m e r a t i o n . S u b s t i t u t i n g 3 ^ 6 ^ g i v e n by E q u a t i o n (7.9) i n t o E q u a t i o n (7.8) and c o n s i d e r i n g t h a t the i n t e g r a l s h o u l d be p e r f o r m e d f o r i from 1 t o 3 we may w r i t e E q u a t i o n (7.8) i n m a t r i x form as U K X ] + [K ] ) < 3 > ( E ) = {K ) 3x3 3x3 1x3 1x3 (a) where i j 3L. 9L 3L 3L. . { - 1 + ( 1 + 2VL.) 1) d D 1 6 ' (b) (e) 3 P 3 P 3C 1 3C (7.10) H i j ° (e) L.L . dS i J (e) ( c ) K = r h ( z ) B ( z ) A. . o a i j L . dS (e) 1 (e) (d) The e v a l u a t i o n o f [K„] , [K„] and [K.] i s g i v e n i n A p p e n d i x I I I . I n A [ K T ] i s c a l c u l a t e d e x a c t l y , [K H ] and l K A 3 a r e c a l c u l a t e d a p p r o x i m a t e l y f o r t h r e e d i f f e r e n t p e r i p h e r i a l e l e m e n t s a c c o r d i n g t o t h e i r o r i e n t a t i o n on t h e e x t e r n a l s u r f a c e w i t h r e s p e c t t o t h e a x i s o f t h e c r y s t a l ; i ) v e r t i c a l , i i ) t i l t e d and i i i ) h o r i z o n t a l p e r i p h e r i a l e l e m e n t s . T r p * e ) TTVb. , . K = ( b . b . + c . c . ) + 1 [3 p ( e ) + p.] T i j 2 A 1 J 1 > 6 J 52 where i , j = 1,2,3 p, * P 2 * P 3 (1) V e r t i c a l b o undary [ K H ] = 2 T r h ( z ) r ( ) P l l j 1/3 1/6 0 1/6 1/3 0 0 0 0 <K } = T r h ( z ) r B ( z ) p l A 0 a 1 j 1 1 >• 0 (7.11) ( i 1 ) T i l t e d Boundary 27Th(z)r ( hi 12 3P. + p. P. + P. P. + P. P. + 3 p. 0 0 0 T T h ( z ) r 0 B a ( z ) (pi + 2 p ) Al 1 1 0 53 ( H i ) H o r i z o n t a l b oundary — ( e) The same as i n ( i ) but r e p l a c i n g p b y p v ' 1 ^ i s t h e l e n g t h of t h e e l e m e n t at the b o u n d a r y . D e t a i l s o f t h e n u m e r i c a l c a l c u l a t i o n s i n c l u d i n g computer c o d i n g , t y p e o f mesh, mesh r e f i n e m e n t , m a t r i x a s s e m b l y and m a t r i x c a l c u l a t i o n s a r e g i v e n l a t e r . 7.1.2 S t r e s s F i e l d Model To c a l c u l a t e t h e t h e r m a l s t r e s s e s i n t h e c r y s t a l t h e f o l l o w i n g a s s u m p t i o n s a r e made : i . t h e body i s l i n e a r l y e l a s t i c and i s o t r o p i c i i . t h e t h e r m o e l a s t i c f i e l d i s a x i s y m m e t r i c i i i . t h e c r y s t a l i s f r e e from t r a c t i o n a t t h e s u r f a c e , i . e . a t the s u r f a c e o x n- = 0 , n i s a v e c t o r p e r p e n d i c u l a r t o t h e s u r f a c e of t h e c r y s t a l . 7.1.2.1 G o v e r n i n g and Element E q u a t i o n s To o b t a i n t h e el e m e n t e q u a t i o n , t h e minimum p o t e n t i a l e n e r g y p r i n c i p l e was u s e d . The s t r a i n e n e r g y f o r a l i n e a r e l a s t i c s o l i d i s d e f i n e d as U (u,w) = 1 p 2 [e] {a} dv (7.12) V where (a) i s t h e s t r e s s t e n s o r and [e] i s t h e s t r a i n t e n s o r w h i c h f o r t h e p r e s e n t c a s e have t h e f o l l o w i n g components 54 { o> TZ (7.13) The s t r e s s , a c c o r d i n g t o Hook's Law and t h e Duhame1-Newman c o n s t i t u t i v e e q u a t i o n t h a t a c c o u n t s f o r t h e t h e r m a l s t r a i n , i s io) [C] ({e} - {e 0>) (7.14) where [EQ] i s t h e s t r a i n c o r r e s p o n d i n g t o a f r e e e x p a n s i o n o f t h e s o l i d due to t e m p e r a t u r e ' d i f f e r e n c e s w i t h r e s p e c t t o r e f e r e n c e t e m p e r a t u r e T Q . T h i s s t r a i n f o r an i s o t r o p i c m a t e r i a l w i t h a c o e f f i c i e n t o f t h e r m a l e x p a n s i o n a i s a( T V a (7.15) 1 0 [C] i s t h e c o m p l i a n c e mat r i x whi c h f o r t h i s c a s e 1- V V V 0 E V 1-V V 0 [C] — V V 1- V 0 (1+ V ) (1-2V) 0 0 0 1 -2v 2 (7 . 16) w i t h E t h e Young modulus and v t h e P o i s s o n ' s r a t i o . The { e) s t r a i n can be d e r i v e d from t h e d i s p l a c e m e n t {d} as 55 ie ) (7.17) where [B] i s t h e f o l l o w i n g m a t r i x of d e r i v a t i v e s [B] 3r J r 0 _3 3z~ "3~i~ 3 3r (7.18) T a k i n g i n t o a c c o u n t t h e r e l a t i o n among s t r e s s , s t r a i n and d i s p l a c e m e n t , t h e p o t e n t i a l e n e r g y can be w r i t t e n as II(u,w) = U (u,w) [d] [B] [C] [B] {d} 2 [ d ] [B] [C] {eQ} dV (7.19) When t h e body i s s u b d i v i d e d i n t o s m a l l e l e m e n t s , i t i s assumed t h a t w i t h i n each e l e m e n t w i t h g nodes t h e d i s p l a c e m e n t f i e l d i s a p p r o x i m a t e d by <;>,e) = te!},e' = { i ^ O = [ N ] , d , (e) (7 . 20) 56 where [N] a r e t h e i n t e r p o l a t i o n f u n c t i o n s m a t r i x of o r d e r g and (e ) {d} a r e t h e e l e m e n t n o d a l d i s p l a c e m e n t s . In t h i s c a s e t h e m i n i m i z a t i o n p r o c e s s may be c a r r i e d out e l e m e n t by e l e m e n t . As a r e s u l t t h e f o r c e - d i s p l a c e m e n t e q u a t i o n f o r t h e e l e m e n t i s o b t a i n e d as 2gx2g 2 g x l 2 g x l [ k ] ( e ) { d } ( 6 ) = { F } ( 6 ) {d} (e) {d> {d}, (e) (e) w. w. {d} (e) g w g (7.21) 2gx2g [k] (e) 2 g x l ( F ) (e) [k] [k] [k] (e) 11 (e) i l (e) g l {F> {F} { F } (e) 1 (e) 2 (e) g ™[V ^ [ k l j ^ t k] t k ] ^ ^ [k] (e) In (e) i g (e) gg (7.22) 57 In t h e s t i f f n e s s m a t r i x [ K ] , t h e s u b m a t r i c e s [ k ] ^ j a r e t h e s t i f f n e s s m a t r i c e s a t node i and have th e form [k] (e) i j (e) 2 x 4 m / x 4x4. . 4x2. . [ B ] { ( e ) [C]< e> [B]< e> dV [B] (e) 3N 3r N 3N 3z 3N 3z 3N , 3r (7.23) The i n i t i a l f o r c e m a t r i x { F Q ^ (e) l s [ F } ( C )  1 O' i (e) 2x4 . . 4 x 4 4x1 . . [B]\(e) [C] f e 0 ] { e , dV (7.24) 7.1.2.2 R e q u i r e m e n t s f o r t h e i n t e r p o l a t i o n f u n c t i o n s . S i n c e o n l y f i r s t o r d e r d e r i v a t i v e s o f d i s p l a c e m e n t a p p e a r i n t h e i n t e g r a n d f o r the p o t e n t i a l e n e r g y , c o m p a t i b i l i t y r e q u i r e s an i n t e r p o l a t i o n f u n c t i o n w i t h c o n t i n u i t y o f d i s p l a c e m e n t a t t h e 58 e l e m e n t b o u n d a r y o n l y . However, f o r c o m p l e t e n e s s , th e I n t e r p o l a t i o n f u n c t i o n s must be a b l e t o r e p r e s e n t r i g i d body d i s p l a c e m e n t s and c o n s t a n t s t r a i n s t a t e s w i t h i n an e l e m e n t ; t h a t i s u n i f o r m s t a t e s of d i s p l a c e m e n t and f i r s t d e r i v a t i v e s o f d i s p l a c e m e n t s r e s p e c t i v e l y . T h i s C^ 1^ c o n t i n u i t y i s s a t i s f i e d w i t h i n t e r p o l a t i o n f u n c t i o n s c o n t a i n i n g l i n e a r terms s u c h as t h e n a t u r a l c o o r d i n a t e s used f o r t h e t e m p e r a t u r e f i e l d . T h i s means t h a t e l e m e n t s w h i c h a r e l i n e a r or o f h i g h e r o r d e r a r e s u i t a b l e . F o r c o n v e n i e n c e , b e f o r e o b t a i n i n g t h e s p e c i f i c form o f e l e m e n t m a t r i c e s , t h e node e q u a t i o n i s w r i t t e n i n terms o f non- d i m e n s i o n a l q u a n t i t i e s as 2x2 . . 2x1 . 2x1 . . f k ' l i j { d ' / j = <V>i ( a ) (7.25) where E { d . } ( e ) = { d } ( e ) ( f e ) a{ T - T ) r MP 0' l0 ( e ) { k ' ) ! ^ = / [ B ' ] j [ C ] [ B ' ] J 6 ) PdPdC (c) where [ B ' ] < e ) = r Q [B][e) (a) ( 1 + V ) ( 1 - 2 V ) [ C ] = [C] (b) E 59 { F - ><e> i r O ' l T ( e ) 6d / [ B 1 ]J [C')< i ( 7.26) V pd pd C ( c ) 7.1.2.3 F o r m u l a s f o r S t r e s s C a l c u l a t i o n s The s t r e s s t e n s o r i s c a l c u l a t e d a t e a c h node i i n e l e m e n t e u s i n g t h e f o l l o w i n g e q u a t i o n ,(e) o. o. > - 1- V V V V 1 - V V v v 1-v o 0 1-2 V > - < 6.. 1 V (7.27) The s t r a i n components a t e ach node a r e c a l c u l a t e d from t h e n o d a l d i s p l a c e m e n t s as ie'}{e) - [ B 1 ] ! 6 ) < d ' ) ! e ) (7.28) F o r nodes a t t h e a x i s o f the body e Q i s s u b s t i t u t e d f o r e . 0 p The s t r e s s c a l c u l a t e d i n t h i s way i s n o n - d i m e n s i o n a l . To o b t a i n t h e s t r e s s i n a b s o l u t e u n i t s t h e f o l l o w i n g r e l a t i o n i s e mployed 60 {0> (e) E a (T MP (1-2V) T ) (7 . 29) 7.1.2.4 L i n e a r E l e m e n t s The l i n e a r e l e m e n t s a r e t r i a n g u l a r t o r o i d s w i t h nodes a t t h e v e r t i c e s o f t h e t r i a n g l e . The i n t e r p o l a t i o n f u n c t i o n s f o r t h e s e e l e m e n t s a r e t h e n a t u r a l c o o r d i n a t e s L , L 0 and L and t h e d i s p l a c e m e n t i s w r i t t e n as {d} (e) L 1 U 1 + L 2 U 2 + L 3 U 3 L 1 W 1 + L 2 W 2 + L 3 W 3 . (7.30) Mak i n g t h e c o r r e s p o n d i n g c a l c u l a t i o n s and s u b s t i t u t i o n s , t h e terms of t h e s t i f f n e s s and i n i t i a l f o r c e m a t r i c e s a r e [ k , , i j } 11 12 21 22 (a) 11 (1-2 V) P v [(1-v) b.b. + I j c.c ] — + - [b.+b.] + 2 J 4A 6 J + ( I - V ) A ;- P dpd? (b) 12 ( 1 -2 v) [ vb . c . + i J c i V P v 4 A 6 + c . J (c) 61 21 ( l - 2 v ) p V [ vb c + c b ] + c (d) J 1 2 J 1 4A 6 22 [ ( l - V ) C ^ C j + ( l - 2 v ) b l b j ] (e) 4A (7.31 ) 2 A ( b j p + ( f ) F ' (g) An I n s p e c t i o n o f t h e above r e l a t i o n s shows t h a t a l l t h e terms can be e v a l u a t e d e x a c t l y e x c e p t th e term c o n t a i n i n g 1/p w h i c h s h o u l d be i n t e g r a t e d n u m e r i c a l l y . One i n t e r m e d i a t e a p p r o x i m a t i o n c o n s i s t s of t a k i n g a v e r a g e v a l u e s of p and (~p and L = 1/3) f o r t h a t term o n l y . Once the d i s p l a c e m e n t f i e l d i s c a l c u l a t e d t h e n o d a l s t r a i n and s t r e s s e s a t e a c h e l e m e n t can be d e r i v e d by back s u b s t i t u t i o n i n t h e c o r r e s p o n d i n g e q u a t i o n s . The s t r e s s e s a t t h e c e n t r o i d of t h e e l e m e n t a r e t h e a v e r a g e s t r e s s e s . The s t r e s s a s s o c i a t e d w i t h e a c h node i s t h e a v e r a g e v a l u e o f s t r e s s e s from a l l e l e m e n t s s h a r i n g t h e node. 7.1.2.5 Q u a d r a t i c E l e m e n t The q u a d r a t i c e l e m e n t has s i x nodes, t h r e e nodes at t h e v e r t i c e s and t h r e e nodes a t m l d s i d e s . I n t e r p o l a t i o n f u n c t i o n s a r e 62 b u i l t from t h e n a t u r a l c o o r d i n a t e s . A d e t a i l e d c a l c u l a t i o n o f s t i f f n e s s and f o r c e m a t r i x e l e m e n t s i s g i v e n i n A p p e n d i x V. I t i s i m p o r t a n t t o n o t e t h a t the n o d a l [B] m a t r i x of t h e s e e l e m e n t s i n c l u d e s o n l y d e r i v a t i v e s of t h e n o n - z e r o t e r m s o f d i s p l a c e m e n t a t t h a t node i n s t e a d o f t h e d e r i v a t i v e o f a l l the terms i n the i n t e r p o l a t e d d i s p l a c e m e n t i n t h e e l e m e n t . More s p e c i f i c a l l y , i n the e l e m e n t t h e d i s p l a c e m e n t components a r e i n t e r p o l a t e d as u s u a l as i \ 6 i \ 6 u i e ; = Z N.u. w*6' = £ N.w, ( 7 . 3 2 ) j = i 3 3 j = l J J where U j and w.. a r e t h e n o d a l d i s p l a c e m e n t s and N t h e i n t e r p o l a t i o n f u n c t i o n s . F o r i n s t a n c e , by d e f i n i t i o n , . 3 u ( e ) 6 3N. e l e ; = = Z J — u ( 7 . 3 3 ) 3r j = l 9r 3 and a t r u e q u a d r a t i c element w i l l have a B el e m e n t a t node i which i s 6 3N. B 1 1 ( £ — N.) ( 7 . 3 4 ) i j = l 3r J N, = 0 N J . - 1 J * 1 However i n t h e p s e u d o - q u a d r a t i c e l e m e n t s employed h e r e , o n l y t h e term N^. u^ i s c o n s i d e r e d , i . e . 3N B = U l ( 7 . 3 5 ) i or 63 I t i s a l s o i m p o r t a n t t o n o t e t h a t f o r t h e c a l c u l a t i o n of s t r e s s e s t h e f u l l [B] m a t r i x i s used i n o r d e r t o o b t a i n r e p r e s e n t a t i v e and c o n s i s t e n t v a l u e s of s t r e s s e s . 7.1.3 Von M i s e s and R e s o l v e d S h e a r S t r e s s e s The s t r e s s t e n s o r i n c y l i n d r i c a l c o o r d i n a t e s o b t a i n e d w i t h t h e f i n i t e e l e m e n t method g i v e s a q u a n t i t a t i v e d e s c r i p t i o n o f t h e t h e r m a l l y i n d u c e d s t r e s s e s . F o r a d i s l o c a t i o n g e n e r a t i o n a n a l y s i s , t h e s e s t r e s s components s h o u l d be t r a n s f o r m e d i n t o q u a n t i t i e s t h a t can be a s s o c i a t e d w i t h the p l a s t i c d e f o r m a t i o n of t h e m a t e r i a l . T h i s s t r e s s f i e l d can be d e s c r i b e d i n terms o f t h e Von M i s e s and R e s o l v e d S h e a r S t r e s s e s (RSS) and t h e s e v a l u e s must be compared t o t h e y i e l d s t r e s s and C r i t i c a l R e s o l v e d S h e a r S t r e s s f o r GaAs to d e t e r m i n e i f d i s l o c a t i o n s a r e g e n e r a t e d . The Von M i s e s S t r e s s i s r e l a t e d t o the p r i n c i p a l s t r e s s e s by G V M = «J 1/2 [ < 0 l ' 0 2 ) 2 + (a1 ~ a 3 ) 2 + ( o 2 - o 3 ) 2 ] (7.36) The p r i n c i p a l s t r e s s e s , by d e f i n i t i o n , a r e the v a l u e s o f the s t r e s s e s t h a t d i a g o n a l i z e t h e s t r e s s t e n s o r . The R S S 1 s are o b t a i n e d by r e s o l v i n g the l o c a l s t r e s s t e n s o r on t h e s l i p p l a n e s and s l i p d i r e c t i o n s . In GaAs the g l i d e s y s t e m i s f o r m e d by the {111} p l a n e s and <110> d i r e c t i o n s g i v i n g 12 s l i p c o m b i n a t i o n s . The p r o c e d u r e as w e l l as the d e t a i l e d c a l c u l a t i o n t o o b t a i n t h e t w e l v e RSS components i s g i v e n i n A p p e n d i x V I . 64 The t w e l v e RSS f o r a c r y s t a l grown i n the [001] d i r e c t i o n a r e g i v e n i n T a b l e 7.2 I t can be s e e n t h a t i n d e p e n d e n t of t h e s t r e s s l e v e l , at e v e r y p o i n t i n s p a c e t h e r e a r e t h r e e c o m b i n a t i o n s of p l a n e s and d i r e c t i o n s t h a t have th e same r e s o l v e d s h e a r s t r e s s , t h e s e a r e (111) [110] ( T i l ) [ H O ] ( T i l ) [110] T h i s l e a v e s o n l y t e n d i f f e r e n t s t r e s s l e v e l s w h i c h w i l l be c a l l e d " g l i d e modes". The d i f f e r e n c e between modes l a b e l l e d a and b i s t h e s i g n of t h e term i n v o l v i n g t h e s h e a r s t r e s s . The t e n modes a r e l i s t e d i n T a b l e 7.1 T h r e e d i m e n s i o n a l s t r e s s c o n t o u r s a r e d i f f i c u l t t o p r o d u c e , t h e r e f o r e f o r t h e a n a l y s i s of s t r e s s d i s t r i b u t i o n two s e c t i o n s o f t h e c r y s t a l a r e used, s e c t i o n s normal and p a r a l l e l t o t h e a x i s or growth d i r e c t i o n . F o r normal s e c t i o n s s t r e s s e s as g i v e n i n T a b l e 7.1 a r e u s e d . F o r p a r a l l e l s e c t i o n s t h e (010) p l a n e i s c h o s e n i n view o f t h e e x p e r i m e n t a l e v i d e n c e t h a t h i g h e r d i s l o c a t i o n d e n s i t i e s o c c u r i n t h a t p l a n e . F o r t h e s e p l a n e s 8 = 0. S u b s t i t u t i n g i n the g i v e n e q u a t i o n s f o r the t e n modes g i v e s the r e s u l t s shown i n T a b l e 7.2 I t can be seen t h a t i n t h i s p l a n e t h e r e a r e o n l y f i v e d i f f e r e n t s t r e s s l e v e l s w i t h the c o r r e s p o n d i n g m u l t i p l i c i t y . The RSS a r e compared w i t h d i f f e r e n t c r i t i c a l s t r e s s e s g i v e n i n t h e l i t e r a t u r e f o r y i e l d , d i s l o c a t i o n g e n e r a t i o n and 6 5 m u l t i p l i c a t i o n f o r doped and undoped GaAs as d e s c r i b e d i n s e c t i o n 8 . 1 . T a b l e 7.1 R e s o l v e d Shear S t r e s s Components i n (001) C r y s t a l s P l a n e | D i r e c t i o n | R e s o l v e d Shear S t r e s s Node [110] / 6 { - ( a - O q ) C O S 2 6 + / 2 T J C O S ( 8 + TT/4 ) } 6 P y per I a ( H I ) [ O i l ] / 6 {-(0 - o 0 ) / 2 s i n e s i n ( 6 + Tr/4) + ( a _ - o 0 ) + ~g" P o t, a + a p£.cos6) I I a [101] /6 {o - a J / 2 c o s 9 s i n ( 9+TT/4 )-( o - o Q ) - P e s 6' - T p ? s i n 6 } I I I a [101] / 6 { ( a - 0 Q ) / 2 c o s 6 c o s ( e + 7 r / 4 ) - ( a r - o a ) ~~g P 0 t, t) - T p C s i n 9 } IV a [ H I ] [ O i l ] / 6 { ( o - o Q ) / 2 s i n 6 c o s ( 9+7T/4 ) + ( o - o j g P 0 C O - T _cos6} PC V b [110] / 6 (- ( a p - o Q ) cos26 + / 2 T p ^ c o s (6 + 7T/4 ) } I a [ O i l ] j/6 { - ( a - O o ) / 2 s i n 9 s i n ( e + Tr/4) + ( o r - a f i ) - 6 P e c e - T ? COS0} I I b (111) [101] / 6 ( ( a -o„)/2cosesin(9+77/4)-(a^-o.) + -g- p 8 C 6 + T p ? s i n 9 } I I I a [110] / 6 {-(o - o r t ) c o s 2 6 + / 2 T C O S ( 9 + T T / 4 ) } g P 6 p ? I a C o n t . / 66 C o n t . / [ O i l ] /6 6 {(a - o Q ) / 2 s i n 6 c o 8 ( 8+TT/4) + ( o_-o Q) + p 0 t, 0 + T r c o s 6} p£ V a (111) [ T o T ] j/6 6 {(a - o J / 2 c o s 6cos ( 6+ n/4 ) - ( a - o.) + P e c e + T . s i n e ) p£ IV b [110] 6 {-( 0 - 0 Q ) C 0 S 2 0-/2 T ,,008(9+^/4)} p o pt, I b 67 T a b l e 7 . 2 R e s o l v e d S h e a r a [001] C r y s t a l S t r e s s Components i n a (010) P l a n e f o r Node | RSS | P r e v i o u s Modes | P l a n e | D i r e c t i o n | 1 A 6 I a (111) [ 110] (111) [ H O ] (111) [ 110] 2 /6 6 I b (111) [ 110] 3 6 11 V a a (111) [ o i l ] (111) [ 011] 4 jM 6 [ ( O - O J - T ] 11 V b b (111) [ O i l ] (111) [ O i l ] 5 6 [ o - o J P K 111 I I I IV IV a b a b (111) [ 101] (TTi) [ToT] (Tii) [ToT] ( i l l ) [10T] 7.2 M o d e l l i n g f o r C o o l i n g A f t e r growth Once t h e c r y s t a l has r e a c h e d i t s f i n a l l e n g t h , i t i s p u l l e d up to s e p a r a t e i t from the melt and i n i t i a t e c o o l i n g to room t e m p e r a t u r e . The e f f e c t o f c o o l i n g i n g e n e r a t i n g d i s l o c a t i o n s has not been e s t a b l i s h e d i n the l i t e r a t u r e . However, t h e c o o l i n g 68 p r a c t i c e may i n d u c e t h e r m a l s t r e s s e s t h a t w i l l g e n e r a t e more d i s l o c a t i o n s , p a r t i c u l a r l y i n t h e hot t a i l end r e g i o n . In o r d e r t o s t u d y the e f f e c t s o f c o o l i n g i n g e n e r a t i n g d i s l o c a t i o n s the f o l l o w i n g m a t h e m a t i c a l a p p r o a c h has been d e v e l o p e d . To o b t a i n t h e t i m e d e p e n d e n t t e m p e r a t u r e f i e l d an a n a l y t i c a l s o l u t i o n was c h o s e n . The s e l e c t i o n o f t h i s method was b a s e d on two f a c t o r s . a) S o l u t i o n s a r e e a s y to o b t a i n and n u m e r i c a l e v a l u a t i o n s a r e cheap compared t o n u m e r i c a l methods, b) The i m p o r t a n c e of c o o l i n g has not been s u f f i c i e n t l y e s t a b l i s h e d t o j u s t i f y the use of e x p e n s i v e and t i m e c o n s u m i n g n u m e r i c a l methods. The a n a l y t i c a l t e m p e r a t u r e f i e l d s were c a l c u l a t e d making t h e f o l l o w i n g a s s u m p t i o n s : 1. The t e m p e r a t u r e f i e l d i s t i m e d e p e n d e n t and a x i s y m m e t r i c . 2. The c r y s t a l i s a f i n i t e c y l i n d e r w i t h c o n s t a n t c r o s s - s e c t i o n . 3. The c r y s t a l i s immersed i n a medium (gas or l i q u i d ) a t a c o n s t a n t t e m p e r a t u r e . 4. Newton's Law of C o o l i n g a p p l i e s a t t h e c r y s t a l s u r f a c e ; the h e a t t r a n s f e r c o e f f i c i e n t i s c o n s t a n t and i t s v a l u e i s d e t e r m i n e d by t h e t e m p e r a t u r e and n a t u r e o f t h e s u r r o u n d i n g medium. 5. The i n i t i a l t e m p e r a t u r e of t h e c r y s t a l i s a x i a l l y d e p e n d e n t ( r a d i a l t e m p e r a t u r e g r a d i e n t s d u r i n g growth a r e 69 s m a l l ) ; and i t i s g i v e n by a p a r a b o l i c f u n c t i o n t h a t f i t s the temperature f i e l d at the a x i s of the c r y s t a l at the moment i t i s p u l l e d from the m e l t . With these assumptions, the temperature f i e l d s s a t i s f i e s the f o l l o w i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n and c o n d i t i o n s u s i n g non- d i m e n s i o n a l q u a n t i t i e s as b e f o r e : 3 3 3 (P ) + 3 p 3 p 3 2 3 3 3 3 t (7.37) where t ( K / r 0 * ) t I n i t i a l C o n d i t i o n at t 6 0 = f ( P ) P 0 + P j S + P 2 C (7.38) Boundary C o n d i t i o n s , w i t h h 1 = hr at t > 0 0 3 3 3 p 3 3 3 C 3 3 3 C -h' 3 P =1 •h' 3 h 1 3 5 =o (a) (b) (c) (7.39) 70 The s o l u t i o n to t h i s p r o b l e m as shown i n A p p e n d i x V i s g i v e n by 2 * 3 ( p . £ . t * ) = 2h'[Z CAX) e~X 1 J . ( A p ) ] X 1 0 2 * {£ C 2 < Y ) e Y 1 [YcosyC + h ' s l n Y q (7.40) Y where t h e \- v a l u e s and y - v a l u e s a r e the s o l u t i o n s o f t h e f o l l o w i n g a l g e b r a i c e q u a t i o n s A J j U ) = h' J 0 ( A ) (a) 2y h' (7.41) t a n Y C t = — -j- (b) y - h 1 w i t h C j < M = ~2 ~2 (a) (h'd + A ) J 0 ( A ) and 11 C 2 ( Y ) - (b) II = s i n y ^ { p Q + P j ( C t + h ' / Y 2 ) + P 2 ( C 2 - 2 / Y 2 ) 2 C O S y h + 2 C t h ' / Y } + <-P 0 h' + p l ( 1 " h ' C t ) 2 2 P 0 h ' P1 2 p ? n ' + P 2 ( 2 ? t " h ' C t + 2 n ' / Y 2 } + - £ — - - 1 - — f - ( c ) Y Y Y 71 €t , 2 L , 2 , ( Y + h' ) + h (7.42) (d) The t e m p e r a t u r e f i e l d c a l c u l a t e d a t a g i v e n t i m e i s employed to c a l c u l a t e t h e t h e r m a l l y i n d u c e d s t r e s s f i e l d , w i t h t h e a i d of t h e f i n i t e e l e m e n t method d e s c r i b e d i n s e c t i o n 7.1 . From t h e s e s t r e s s e s t h e Von M i s e s and RSS a r e d e r i v e d and compared t o t e m p e r a t u r e d e p e n d e n t c r i t i c a l v a l u e s f o r y i e l d or d i s l o c a t i o n g e n e r a t i on. 7.3 A n a l y t i c a l S o l u t i o n s A n a l y t i c a l s o l u t i o n s f o r t h e t e m p e r a t u r e and s t r e s s f i e l d s were o b t a i n e d and n u m e r i c a l e v a l u a t i o n s o f t h e s e s o l u t i o n s were compared w i t h t h o s e o b t a i n e d w i t h f i n i t e e l e m e n t methods. The a n a l y t i c a l s o l u t i o n s d e v e l o p e d were o b t a i n e d f o r s i m p l i f i e d b o u n d a r y c o n d i t i o n s . 7.3.1 A n a l y t i c a l QSS T e m p e r a t u r e F i e l d The q u a s i - s t e a d y s t a t e d i f f e r e n t i a l e q u a t i o n (7.1) i s s o l v e d a n a l y t i c a l l y f o r a s i m p l i f i e d p r o b l e m i n which the f o l l o w i n g c o n d i t i o n s a r e a d o p t e d : 1. P l a n a r s o l i d - l i q u i d i n t e r f a c e 2. C y l i n d r i c a l c r y s t a l w i t h c o n s t a n t c r o s s - s e c t i o n 72 3. U n i f o r m h e a t t r a n s f e r c o e f f i c i e n t h and c o n s t a n t ambient t e m p e r a t u r e T f l 4. C o n s t a n t t e m p e r a t u r e a t t h e ends o f t h e c r y s t a l From t h e s e , t h e b o u n d a r y c o n d i t i o n s a r e w r i t t e n as 9 3 " 9 p p=l h 1 3 s (a) 31 = 1 (b) (7.43) <5=0 31 = 0 ; E = ^- ( c ) 0 where i s t h e t o t a l l e n g t h o f t h e c r y s t a l . The s o l u t i o n to t h e h e a t c o n d u c t i o n e q u a t i o n ( 7 . 1 ) , s u b j e c t t o t h e s p e c i f i e d b o u n d a r y c o n d i t i o n s , i s g i v e n by t h e f o l l o w i n g e q u a t i o n i n term of the B e s s e l f u n c t i o n o f z e r o and f i r s t o r d e r J Q and J j - k x C - 2 k x C T k x C v£ J 0 ( ^P) te - e e g ( p . O = 2hr e v ^ Z ; ] (7.44) A ( A +h r / ) J n ( A ) -2k, c_ 0 0 (1 - e A M where the v a l u e s a r e t h e s o l u t i o n s of the a l g e b r a i c e q u a t i o n A J j ( A ) = h r 0 J Q ( A ) (7.45) and k 2 . v 2 + x 2 A D e t a i l s of the c a l c u a t i o n s a r e g i v e n i n A p p e n d i x V I I 73 (7.46) 7.3.2 A n a l y t i c a l s o l u t i o n s f o r t h e S t r e s s F i e l d A n a l y t i c a l s o l u t i o n s f o r t h e s t r e s s f i e l d were o b t a i n e d u s i n g two d i f f e r e n t a p p r o x i m a t i o n s : p l a n e s t r a i n and a x i s y m m e t r i c . Two t y p e s o f t e m p e r a t u r e f i e l d s were employed, t e m p e r a t u r e f i e l d s t h a t depend on r a d i u s o n l y (no a x i a l t e m p e r a t u r e g r a d i e n t ) and g e n e r a l a x i s y m m e t r i c t e m p e r a t u r e f i e l d s . 7.3.2.1 P l a n e S t r a i n A p p r o x i m a t i o n In t h i s a p p r o x i m a t i o n i t i s assumed t h a t the d i s p l a c e m e n t f i e l d c o n s i s t s o f p r i m a r i l y a r a d i a l component, w i t h t h e a x i a l component c o n s t r a i n e d to m a i n t a i n z e r o a x i a l t r a c t i o n a t t h e two ends of a c y l i n d e r . F o r t h e two d i m e n s i o n a l a x i s y m m e t r i c t e m p e r a t u r e f i e l d c a l c u l a t e d i n s e c t i o n 7.3.1, the t h r e e components of n o n - d i m e n s i o n a l s t r e s s from A p p e n d i x IX a r e 2hr hr Jj(Xp) 1 ] (a) o P Xp J 0 ( A ) 2hr ( Jj(X P) - J 0(XP))] (b) o e Xp o 2hr (c) 74 where (7.47) " k ^ - 2 k A * t k A 5 e - e e K x (C) = (d) (1 - e A l ) ( \ 2 + h 2 r Q 2 ) F o r a r a d i a l t e m p e r a t u r e f i e l d g i v e n by 3 = - P 2 (7.48) the s t r e s s e s a r e g i v e n by the f o l l o w i n g r e l a t i o n s h i p s 0 p = 1/4 ( p 2 - 1) (a) O Q = 1/4 ( 3 p 2 - 1) (b) (7.49) 0 ? = p 2 - 1/2 ( c ) D e t a i l s of the c a l c u l a t i o n s a r e g i v e n i n A p p e n d i x V I I . 7.3.2.2 A x i s y m m e t r i c S o l u t i o n s A n a l y t i c a l s o l u t i o n s u s i n g t h e a x i s y m m e t r i c a p p r o x i m a t i o n were o b t a i n e d u s i n g L o v e ' s and G o o d i e r ' s p o t e n t i a l s . G o o d i e r ' s p o t e n t i a l g i v e s a p a r t i c u l a r s o l u t i o n f o r t h e t h e r m o e l a s t i c s t r e s s e s w h i l e L o v e ' s p o t e n t i a l g i v e s a g e n e r a l s o l u t i o n f o r t h e i s o t h e r m a l e l a s t i c s t r e s s e s . A c o m b i n a t i o n of b o t h p o t e n t i a l s i s n e c e s s a r y t o s a t i s f y the b o u n d a r y c o n d i t i o n s . In t h i s c a s e t h e body i s a c y l i n d e r w i t h c o n s t a n t c r o s s - s e c t i o n and f i n i t e l e n g t h . S o l u t i o n s a r e o b t a i n e d u s i n g f i n i t e F o u r i e r t r a n s f o r m s f o r 75 a x i s y m m e t r i c and r a d i a l t e m p e r a t u r e f i e l d s as i n the p l a i n s t r a i n c a s e . D e t a i l s o f t h e c a l c u l a t i o n s and s o l u t i o n s a r e g i v e n i n A p p e n d i x IX. F o r p u r p o s e s o f c o m p a r i s o n w i t h t h e f i n i t e e l e m e n t method, n o n - d i m e n s i o n a l s t r e s s component were o b t a i n e d by m u l t i p l y i n g t h e n o n - d i m e n s i o n a l s t r e s s e s by, a) 1 - 2v f o r t h e a x i s y m m e t r i c a p p r o x i m a t i o n and b) (1 - 2v) / (1 - v ) f o r t h e p l a n e s t r a i n a p p r o x i m a t i o n w i t h v = 0.29 76 CHAPTER 8 EVALUATION OF THE MODELS In t h i s c h a p t e r the r e s u l t s o f t h e models a r e compared w i t h t h e a n a l y t i c a l s o l u t i o n s p r e s e n t e d i n t h e l a s t c h a p t e r . In a d d i t i o n the t e m p e r a t u r e f i e l d s o b t a i n e d w i t h t h e f i n i t e e l e m e n t method a r e compared w i t h t e m p e r a t u r e measurements i n a h i g h p r e s s u r e M e l b o u r n p u l l e r d u r i n g g r o w t h . The model i s e v a l u a t e d on t h e b a s i s of t h e s e r e s u l t s . 8.1 Programming and I n p u t P a r a m e t e r s B ased on t h e n u m e r i c a l scheme p r e s e n t e d i n t h e l a s t c h a p t e r , programs i n FORTRAN l a n g u a g e were w r i t t e n f o r the c a l c u l a t i o n o f t e m p e r a t u r e s and s t r e s s e s i n t h e GaAs c r y s t a l d u r i n g g r o w t h . Flow c h a r t s a r e shown i n F i g u r e s 8.1 and 8.2 The f l o w c h a r t s c o r r e s p o n d i n g to t h e computer programs employed f o r c a l c u l a t i n g the s t r e s s e s w i t h l i n e a r and q u a d r a t i c e l e m e n t s a r e e s s e n t i a l l y t h e same. The mesh was a u t o m a t i c a l l y g e n e r a t e d u s i n g a computer program w i t h a f l o w c h a r t as shown i n F i g u r e 8.3. S e v e r a l d i f f e r e n t meshes u s i n g t r i a n g u l a r e l e m e n t s were t e s t e d i n o r d e r to s e l e c t the one which gave the b e s t r e s u l t s . A C h o l e s k i method was s e l e c t e d f o r s o l v i n g t h e l i n e a r s y s t e m of e q u a t i o n s t h a t r e s u l t e d from t h e f i n i t e e l e m e n t method. In t h e c a s e o f the t e m p e r a t u r e f i e l d c a l c u l a t i o n s , t h e t e s t employed f o r the s e l e c t i o n o f t h e s o l u t i o n method and optimum mesh 77 ( START thermal d l f f u s i v i t y temperature p r o f i l e boron oxide t h i c k n e s s pressure growth v e l o c i t y c r y s t a l geometry : r Q , z , cone angle, seed l e n g t h , seed l e n g t h , seed diameter. •esh data : number of nodes, elements, nodes with constant temperature ( i n t e r f a c e ) , boundary segments ( v e r t i c a l , t i l t e d , h o r i z o n t a l ) nodal c o o r d i n a t e s , system topology. I N I T I A L I Z E : [ K T ] , [ K H ] , [ K A ] , 0 For each element EVALUATE : [ k T ] e ASSEMBLE g l o b a l matrix [ ] without r e g a r d i n g boundary c o n d i t i o n s 78 F o r e a c h b o u n d a r y e l e m e n t ACCOUNT FOR c o n v e c t i o n - CALCULATE c o r r e s p o n d i n g h e a t t r a n s f e r c o e f f i c i e n t and a m b i e n t t e m p e r a t u r e - EVALUATE [ K H ] and [ K A ] - ADD [ K T ] and [ K H ] F o r e a c h node w i t h s p e c i f i e d MODIFY [ K A ] and [ K ^ ] t e m p e r a t u r e SOLVE t h e l i n e a r s y s t e m [K ] { g } = [K ] u s i n g C h o l e s k i ' s method. A PRINT t e m p e r a t u r e s t o be r e a d by PFEMS ^ PLOT t e m p e r a t u r e c o n t o u r s STOP F i g u r e 8.1 F l o w c h a r t o f t h e c o m p u t e r p r o g r a m t o c a l c u l a t e t h e t e m p e r a t u r e f i e l d i n t h e c r y s t a l u s i n g a f i n i t e e l e m e n t method. ( START ") INPUT P o i s s o n r a t i o c r y s t a l d i m e n s i o n s ; r , Z • e s h d a t a : number o f nodes, e l e m e n t s , nodes w i t h s p e c i f i e d d i s p l a c e m e n t , n o d a l s c o o r d i n a t e s , s y s t e m t o p o l o g y n o d a l t e m p e r a t u r e s I N I T I A L I Z E : [K'] , {F^} F o r e a c h e l e m e n t , a t e a c h node (e) EVALUATE FORM [k 1 ] <F' } (e) i j * ' i u s i n g q u i n t i c o r d e r i n t e g r a t i o n m a t r i c e s [ k ' ] e and ( F ^ } e i n e l e m e n t ASSEMBLE G l o b a l M a t r i c e s [K'] and {F^} w i t h o u t r e g a r d i n g nodes w i t h s p e c i f i e d d i s p l a c e m e n t 5 80 2 ACCOUNT f o r nodes w i t h s p e c i f i e d d i s p l a c e m e n t SOLVE OBTAIN t h e l i n e a r s y s t e m [ K 1 ] {d 1} u s i n g C h o l e s k i ' s method and d i s p l a c e m e n t s u, w <F 0} Fo r e a c h e l e m e n t a t e a c h node : CALCULATE s t r a i n components e , e „ , e , Y P e c pc CALCULATE s t r e s s components o , o„, o , T P e C P C . At e a c h node CALCULATE a v e r a g e s t r e s s c a l c u l a t e d f r o m a l l e l e m e n t s s h a r i n g the node CALCULATE p r i n c i p a l s t r e s s e s o , 0 , o J. O CALCULATE Von M i s e s s t r e s s 0 V Fl 81 PRINT Output S t r e s s components and Von Mises s t r e s s at a l l nodes PLOT Von Mises s t r e s s contours STOP Fi g u r e 8 . 2 Flow chart of the computer program f o r c a l c u l a t i o n s u s i n g f i n i t e l i n e a r and elements. the s t r e s s q u a d r a t i c 82 ( START ) / ^ I N P U T : e l e m e n t s i z e , c r y s t a l l e n g t h , s e e d w i d t h , s e e d l e n g t h , cone a n g l e I N I T I A L I Z E : l a s t node and l a s t e l e m e n t c a l c u l a t e d LAST, LASTEL. CALCULATE : number o f nodes i n r a d i a l and a x i a l d i r e c t i o n s r e c a l c u l a t e e l e m e n t s i z e f o r t h e b u l k c r y s t a l GENERATE f i r s t row o f nodes a t t h e i n t e r f a c e F o r e a c h a x i a l p o s i t i o n i n t h e b u l k c r y s t a l , GENERATE nodes i n t h e r a d i a l d i r e c t i o n 83 2 F o r e a c h e l e i e n t i n t h e b u l k c r y s t a l , GENERATE b u l k c r y s t a l t o p o l o g y CALCULATE number and d i m e n s i o n o f b l o c k s i n t h e cone F o r e a c h a x i a l p o s i t i o n i n t h e c o n e , CALCULATE number o f nodes i n t h e r a d i a l d i r e c t i o n CALCULATE e l e m e n t s i z e I n r a d i a l d i r e c t i o n GENERATE no d e s i n r a d i a l d i r e c t i o n s F o r e a c h e l e m e n t i n t h e c o n e , GENERATE cone t o p o l o g y CALCULATE number o f no d e s i n r a d i a l and a x i a l d i r e c t i o n s I n t h e s e e d 5 84 © F o r e a c h a x i a l p o s i t i o n i n t h e s e e d , GENERATE nodes i n t h e r a d i a l d i r e c t i o n F o r e a c h e l e m e n t i n t h e s e e d , GENERATE s e e d t o p o l o g y DETERMINE s e g m e n t s where c o n v e c t i o n o c c u r s , - v e r t i c a l s e g m e n t s - t i l t e d s e g m e n t s - h o r i z o n t a l s e g m e n t s / P R I N T D a t a t o be r e a d by PFEMT and PFEMS / STOP F i g u r e 8.3 F l o w c h a r t o f t h e mesh g e n e r a t o r c o m p u t e r p r o g r a m . 85 c o n s i s t e d o f a p p l y i n g u n i f o r m b o u n d a r y c o n d i t i o n s on the s u r f a c e of t h e c r y s t a l . These c o n d i t i o n s were u n i t ambient t e m p e r a t u r e i n t h e medium s u r r o u n d i n g the c r y s t a l , c o n s t a n t h e a t t r a n s f e r c o e f f i c i e n t and u n i t t e m p e r a t u r e a t t h e s o l i d l i q u i d i n t e r f a c e . The t e m p e r a t u r e f i e l d i n the c r y s t a l i n s u c h c o n d i t i o n s s h o u l d be u n i f o r m and have u n i t v a l u e . The c o m b i n a t i o n of mesh t y p e and s o l u t i o n method s e l e c t e d gave t e m p e r a t u r e f i e l d s f o r a c o a r s e mesh, w h i c h were 99 % o f t h e t h e o r e t i c a l v a l u e . In t h e c a s e of t h e t h e r m o e l a s t i c s t r e s s c a l c u l a t i o n s w i t h the FEM, a p p l i c a t i o n of a u n i t t e m p e r a t u r e f i e l d t h r o u g h o u t t h e c r y s t a l gave n o n d i m e n s i o n a l s t r e s s v a l u e s l e s s t h a t 1 0 - 4 . i n s u c h c o n d i t i o n s t h e t h e o r e t i c a l s t r e s s s h o u l d be z e r o . In a d d i t i o n , manual t e s t s were c a r r i e d out u s i n g s i m p l e meshes t o e n s u r e t h a t t h e s t i f f n e s s e l e m e n t m a t r i c e s were s y m m e t r i c and p o s i t i v e l y d e f i n e d ( e i g e n v a l u e s s h o u l d be p o s i t i v e ) . To m i n i m i z e the computer memory r e q u i r e m e n t s , which i s c r i t i c a l f o r l a r g e s y s t e m s , t h e s t i f f n e s s m a t r i x f o r t h e whole s y s t e m was a s s e m b l e d d i r e c t l y i n a l i n e a r a r r a y . The o u t p u t of the c o m p u t a t i o n a l e f f o r t of t h e t e m p e r a t u r e c a l c u l a t i o n s c o n s i s t e d of n o d a l t e m p e r a t u r e s and i s o t h e r m p l o t s i n t h e c r y s t a l . The o u t p u t i n t h e s t r e s s program c o n s i s t e d of n o d a l d i s p l a c e m e n t s , n o d a l s t r e s s components i n c y l i n d r i c a l c o o r d i n a t e s , n o d a l Von M i s e s s t r e s s e s and Von M i s e s i s o s t r e s s c o n t o u r s i n the c r y s t a l . 86 Two o t h e r programs were w r i t t e n f o r c a l c u l a t i n g the maximum component o f r e s o l v e d s h e a r s t r e s s i n p l a n e s p e r p e n d i c u l a r and p a r a l l e l t o t h e c r y s t a l a x i s . These p l a n e s c o r r e s p o n d t o a (001) p l a n e and (010) p l a n e r e s p e c t i v e l y . S i m i l a r p l o t s a r e o b t a i n e d of the d i f f e r e n c e between t h e MRSS and e x p e r i m e n t a l v a l u e s of c r i t i c a l r e s o l v e d s h e a r s t r e s s e s f o r d i s l o c a t i o n g e n e r a t i o n (CRSS) or y i e l d ( Y S ) . V a l u e s o f CRSS f o r doped and Te-doped [ C R S S ( T e ) ] GaAs have been employed f o r t h e c o m p a r i s o n . These v a l u e s a r e t e m p e r a t u r e d e p e n d e n t . The f l o w c h a r t s of the programs a r e shown i n F i g u r e s 8.4 and 8.5. In a l l c a s e s t h e o u t p u t a l s o c o n s i s t s o f n o d a l v a l u e s of the c o r r e s p o n d i n g s t r e s s and t h e mode ( s p e c i f i c p l a n e and d i r e c t i o n ) w h i c h g i v e s t h e maximum component of s t r e s s . The computer programs f o r the f o u r c a l c u l a t i o n s a r e l i s t e d i n A p p e n d i x X I . 8.1.1 I n p u t P a r a m e t e r s The i n p u t p a r a m e t e r s f o r t h e c a l c u l a t i o n s i n t h e d i f f e r e n t computer programs depend on the s t a g e b e i n g m o d e l l e d . F o r t h e mesh g e n e r a t i o n , o n l y g e o m e t r i c p a r a m e t e r s a r e g i v e n , t h e y a r e : 1) c r y s t a l d i m e n s i o n s : - r a d i u s - l e n g t h - cone a n g l e - s e e d l e n g t h - s e e d r a d i u s - i n t e r f a c e shape 2) e l e m e n t s i z e g i v e n by the d i s t a n c e between nodes. / ' I N P U T number o f nodes s t r e s s components n o d a l c o o r d i n a t e s t e m p e r a t u r e s At e a c h node - CALCULATE f i v e components o f t h e r e s o l v e d s h e a r s t r e s s e s In (010) p l a n e s (RSS) - OBTAIN maximum component o f RSS (MRSS) » GET node - SUBSTRACT c o r r e s p o n d i n g c r i t i c a l r e s o l v e d s h e a r s t r e s s (MRSS > - CRSS) PRINT : MRSS - CRSS, mode PLOT : MRSS - CRSS c o n t o u r s ^ S T O P ^ F i g u r e 8.4 Flow c h a r t o f t h e computer program t o c a l c u l a t e and p l o t RSS i n a v e r t i c a l (010) p l a n e INPUT - s t r e s s components i n r a d i a l d i r e c t i o n a t Z - t e m p e r a t u r e s GENERATE p o l a r g r i d At e a c h p o l a r node - CALCULATE t e n components o f t h e r e s o l v e d s h e a r s t r e s s (RSS) - OBTAIN maximum component o f s t r e s s (NRSS) ; GET mode - SUBSTRACT c o r r e s p o n d i n g c r i t i c a l r e s o l v e d s h e a r s t r e s s (MRSS - CRSS) L PRINT : MRSS - CRSS, mode PLOT : MRSS - CRSS c o n t o u r s ( STOP ) F i g u r e 8.4 Flow c h a r t o f t h e computer program t o c a l c u l a t e and p l o t RSS i n a (001) p l a n e 89 The i n p u t f o r the t e m p e r a t u r e program i s the n o d a l and element c o n f i g u r a t i o n , t h e nodes at t h e i n t e r f a c e and t h e segments where he a t i s t r a n s f e r r e d to the s u r r o u n d i n g a t m o s p h e r e . The p h y s i c a l d a t a i n c l u d e s gas p r e s s u r e , b o r i c o x i d e h e i g h t and t e m p e r a t u r e p r o f i l e s i n b o t h media. From t h e s e t h e h e a t t r a n s f e r c o e f f i c i e n t a t t h e c o r r e s p o n d i n g t e m p e r a t u r e i s c a l c u l a t e d i n t h e program. The h e a t t r a n s f e r c o e f f i c i e n t r e l a t i v e t o the t h e r m a l c o n d u c t i v i t y i n c l u d e s h e a t t r a n s f e r by r a d i a t i o n and c o n v e c t i o n h = h + h r c The h e a t t r a n s f e r c o e f f i c i e n t f o r r a d i a t i o n was d e r i v e d by 19 0 191 J o r d a n ' from t h e S t e f a n - B o 1 t z m a n e q u a t i o n l e a d i n g t o the f o l l o w i n g e q u a t i o n : h = (2 . 27 x 1 0 1 1 / K ) £ T 3 r t a where e i s t h e t o t a l e m i t t a n c e o f GaAs and i s a f u n c t i o n of t e m p e r a t u r e and d o p i n g l e v e l . The c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t t o t h e gas and B^Og l a y e r was c a l c u l a t e d a s s u m i n g h e a t i s t r a n s f e r r e d by f r e e 191 c o n v e c t i o n from a v e r t i c a l w a l l i n t o a f l u i d and i s g i v e n by T - T 1 / 4 h rem" 1] = ( -) h" p 1 / 2 C 1 where . 5 4 8 _ J L _ [ p 2 G ^ ^ / K ^ J 1 / 2 K. a 90 h rod t o o o 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 T E M P E R A T U R E ( K ) F i g u r e 8.6 E s t i m a t e d r a d i a t i v e and c o n v e c t i o n h e a t t r a n s f e r c o e f f i c i e n t s f o r GaAs/B 0^ ( 1 ) , He ( g ) , N g (g) and A (g) as a f u n c t i o n or ambient t e m p e r a t u r e . The n u m e r i c a l l a b e l s a r e t h e p r o d u c t of t h e carrier c o n c e n t r a t i o n X c r y s t a l d i a m e t e r i n u n i t s of cm F i g u r e 8.7 T o t a l h e a t t r a n s f e r c o e f f i c i e n t h, i n B 2 ° 3 a n ( * a r £ o n as a f u n c t i o n of t h e ambient t e m p e r a t u r e T (1) T o t a l h e a t t r a n s f e r c o e f f i c i e n t i n B 2 ° 3 ' T o t a l h e a t t r a n s f e r c o e f f i c i e n t i n a r g o n p r e s s u r i z e d a t 30 atm. 91 K = ambient t h e r m a l d i f f u s i v i t y a p = ambient d e n s i t y a a g = c o e f f i c i e n t of t h e r m a l volume e x p a n s i o n Cp = h e a t c a p a c i t y u = v i s c o s i t y a p = p r e s s u r e 1 = h e i g h t of t h e f l u i d column f o r a l i q u i d medium p i s t a k e n as p = 1. The n u m e r i c a l e v a l u a t i o n of t h e h e a t t r a n s f e r c o e f f i c i e n t s 191 was done by J o r d a n e t a l . u s i n g t e m p e r a t u r e dependent v a l u e s o f t h e p h y s i c a l q u a n t i t i e s i n v o l v e d . The r e s u l t s f o r a gas p r e s s u r e of 1 atmosphere a r e shown i n F i g u r e 8.6 f o r t h e d i f f e r e n t media. The t o t a l h e a t t r a n s f e r c o e f f i c i e n t s f o r b o r o n o x i d e and a r g o n at 30 a t m o s p h e r e s a r e shown i n F i g u r e 8.7 In t h e program the heat t r a n s f e r c o e f f i c i e n t s a r e e v a l u a t e d by f u n c t i o n s w h i c h f i t t h e d a t a and i n c l u d e t h e e f f e c t of t e m p e r a t u r e and p r e s s u r e . O t h e r p h y s i c a l p a r a m e t e r s a r e t h e t h e r m a l d i f f u s i v i t y w h i c h i s assumed to be t e m p e r a t u r e i n d e p e n d e n t , and growth v e l o c i t y . The same n o d a l c o n f i g u r a t i o n employed i n t h e c a l c u l a t i o n o f t e m p e r a t u r e s must be used f o r t h e t h e r m a l s t r e s s c a l c u l a t i o n s . The n o d a l t e m p e r a t u r e s a r e t h e i n p u t f o r t h i s program. In a d d i t i o n the P o i s s o n r a t i o and Young modulus a r e g i v e n . The v a l u e s of t h e most i m p o r t a n t p a r a m e t e r s a r e l i s t e d i n T a b l e 8.1. 92 T a b l e 8.1 V a l u e s o f p h y s i c a l p a r a m e t e r s u s e d In t h e c a l c u l a t i o n s M e l t i n g p o i n t T h e r m al c o n d u c t i v i t y T h e r m al d i f f u s i v i t y T h e r m a l e x p a n s i o n c o e f f i c i e n t Young modulus P o i s s o n r a t i o a E / (l - 2 v ) 1238°C 0.08 Watts/cm K 0.04, ± 0.01 (800 C - 1238 C) (cm / s e c ) 1.0 X 1 0 ~ 5 (°K *) 1 . 2 X 1 0 1 1 Pa 0 . 29 2 . 86 ± 0.1 (800 C MPa ° K _ 1 1238 C) The y i e l d and c r i t i c a l s t r e s s e s employed f o r c o m p a r i s o n w i t h the MRSS were m e a s u r e d by Swaminathan and C o p l e y 211 and 16 3 M i l d v i s k i i e t a l . r e s p e c t i v e l y . The y i e l d s t r e s s measured at t e m p e r a t u r e s w e l l below t h e m e l t i n g p o i n t a r e e x t r a p o l a t e d t o t h i s p o i n t ; an i n v e r s e e x p o n e n t i a l t e m p e r a t u r e d e p e n d e n c e was assumed as the e x p e r i m e n t a l r e s u l t s s u g g e s t . The CRSS f o r undoped 10 07 o.) as 1 i 1 1 .L /2V /too 1001 KO t00 It F i g u r e 8.8 T e m p e r a t u r e dependence of t h e c r i t i c a l s t r e s s f o r d i s l o c a t i o n g e n e r a t j g n i g GaAs (1) Te-doped m a t e r i a l , n_= 2 10 " 1 8 7 X4°" 10 cm cm ; (2) T e - d o p e d m a t e r i a l , cm ; (3) Zn-doped m a t e r i a l , p = 9 X (4) undoped m a t e r i a l . 93 and doped GaAs c l o s e t o the m e l t i n g p o i n t a r e shown i n F i g u r e 8.8. I t can be n o t e d t h a t the CRSS f o r doped m a t e r i a l i s about one o r d e r o f m a g n i t u d e l a r g e r t h an f o r undoped m a t e r i a l . 8.1.2 N u m e r i c a l E v a l u a t i o n of the A n a l y t i c a l S o l u t i o n s F o r t h e n u m e r i c a l e v a l u a t i o n of the a n a l y t i c a l s o l u t i o n s f o r t h e d i f f e r e n t f i e l d s and c o n d i t i o n s , programs i n BASIC l a n g u a g e were w r i t t e n . Programs were run i n a VIC-20 m i c r o c o m p u t e r w i t h expanded memory to 8 KRam. The A v a l u e , s o l u t i o n s o f E q u a t i o n (7.45) were o b t a i n e d u s i n g g r a p h i c methods . The f i r s t f i v e r o o t s were use d . The B e s s e l f u n c t i o n J Q , , I Q and 1^ were 213 e v a l u a t e d u s i n g p o l y n o m i a l i n t e r p o l a t i o n s f o r t h e f u n c t i o n s The summation o f t h e F o u r i e r s e r i e s was c a r r i e d out u s i n g between 20 t o 40 t e r m s . T h i s was n e c e s s a r y b e c a u s e o f the t y p i c a l s low c o n v e r g e n c y o f t h i s s e r i e s . 8.2 C o m p a r i s o n of Model P r e d i c t i o n s w i t h A n a l y t i c a l S o l u t i o n s 8.2.1 T e m p e r a t u r e F i e l d N u m e r i c a l s o l u t i o n s o b t a i n e d u s i n g t h e f i n i t e e l ement method were compared w i t h a n a l y t i c a l s o l u t i o n s f o r t h e p r o b l e m d e s c r i b e d i n S e c t i o n 7.3. C a l c u l a t i o n s were p e r f o r m e d f o r two v a l u e s of t h e h e a t t r a n s f e r c o e f f i c i e n t h. The v a l u e s s e l e c t e d , 0.3 cm 1 and 0.6 cm *, a r e i n t h e range a p p l i c a b l e t o a c r y s t a l b e i n g grown i n a p r e s s u r i z e d a r g o n atmosphere of 3.04MPa. The c r y s t a l r a d i u s was 20 mm and t h e h e i g h t 40 mm. The r e s u l t s a r e shown i n F i g u r e s 8.9 94 F i g u r e 8.9 A c o m p a r i s o n of f i n i t e e l e m e n t and a n a l y t i c a l c a l c u l a t e d t e m p e r a t u r e c u r v e s f o r h = 0.3 cm 95 F i g u r e 8.10 and = 0.6 a n a l y t i c a l cm 96 F i g u r e 8.11 Mesh employed i n the c a l c u l a t i o n s of t h e t e m p e r a t u r e f i e l d s shown i n F i g . 8.9 and 8.10. Number o f nodes = 45. Number of e l e m e n t s = 64. 97 and 8.10 from w h i c h I t i s e v i d e n t t h a t t h e agreement i s good. The f i n i t e e l e m e n t method g i v e s t e m p e r a t u r e s w h i c h a r e s l i g h t l y lower than the t e m p e r a t u r e s c a l c u l a t e d a n a l y t i c a l l y . A t o t a l o f 64 e l e m e n t s were employed i n t h e c a l c u l a t i o n and t h e c o r r e s p o n d i n g number of nodes was 45. The mesh used i s shown i n F i g u r e 8.11. F u r t h e r r e f i n e m e n t o f t h e mesh d i d not a p p r e c i a b l y change th e r e s u l t s . 8.2.2 S t r e s s F i e l d s F i n i t e e l e m e n t s o l u t i o n s were o b t a i n e d f o r a c y l i n d r i c a l c r y s t a l f o r r a d i a l and a x i s y m m e t r i c t e m p e r a t u r e f i e l d s u s i n g l i n e a r e l e m e n t s . A c r y s t a l o f r a d i u s 20 mm, and l e n g t h 40 mm was d i s c r e t i z e d i n t o a mesh o f 256 e l e m e n t s w h i c h c o n s i s t s o f 153 nodes. The s t r e s s e s were c a l c u l a t e d on t h e b a s i s o f a) t h e computed n o d a l t e m p e r a t u r e s and b) a u n i f o r m t e m p e r a t u r e i n the element d e r i v e d by a v e r a g i n g t h e n o d a l t e m p e r a t u r e s . F o r r a d i a l t e m p e r a t u r e f i e l d s , the r a d i a l s t r e s s e s c a l c u l a t e d by t h e f i n i t e e l e m e n t method f o r n o d a l and a v e r a g e e l e m e n t t e m p e r a t u r e s a r e p l o t t e d as a f u n c t i o n o f r / r ^ i n F i g u r e 8.12. F o r c o m p a r i s o n t h e c o r r e s p o n d i n g s t r e s s e s d e t e r m i n e d a n a l y t i c a l l y a r e a l s o shown f o r b o t h p l a n e s t r a i n and a x i s y m m e t r i c d i s p l a c e m e n t s . The r a d i a l s t r e s s e s e f f e c t i v e l y c o i n c i d e f o r t h e two a n a l y t i c a l s o l u t i o n s and t h e f i n i t e e l ement s o l u t i o n s u s i n g t h e a v e r a g e e l e m e n t t e m p e r a t u r e . The r e s u l t s f o r t h e f i n i t e e l e m e n t method u s i n g computed n o d a l t e m p e r a t u r e s i s w e l l below th e o t h e r t h r e e . T h i s l a r g e d i f f e r e n c e i s a t t r i b u t e d 98 CO I Q_ i f UJ - 0 2 - 0 4 - - 0 6 - - 0 8 - r/rr F i g u r e 8.12 C a l c u l a t e d r a d i a l s t r e s s e s as a r a d i a l t e m p e r a t u r e f i e l d s (1) a v e r a g e d element (2) F i n i t e t e m p e r a t u r e s (3) A n a l y t i c a l A n a l y t i c a l - a x i s y m m e t r i c . f u n c t i o n o f r / r f o r F i n i t e e l e m e n t w i t h e l e m e n t w i t h n o d a l p l a n e s t r a i n (4) 99 F i g u r e 8.13 C a l c u l a t e d a z i m u t h a l s t r e s s e s as a f u n c t i o n of r / r f o r r a d i a l t e m p e r a t u r e f i e l d s (1) F i n i t e elemen? w i t h a v e r a g e d e l e m e n t t e m p e r a t u r e s (2) F i n i t e e l e m e n t w i t h n o d a l t e m p e r a t u r e s (3) A n a l y t i c a l - p l a n e s t r a i n (4) A n a l y t i c a l - a x i s y m m e t r i c . 100 i 1 1 r -0 -6 0 0 2 0 4 0 6 0 8 r/r. 10 F i g u r e 8.14 C a l c u l a t e d a x i a l s t r e s s e s as a f u n c t i o n o f r / r f o r r a d i a l t e m p e r a t u r e f i e l d s (1) F i n i t e e l e m e n t w i t h a v e r a g e d e l e m e n t t e m p e r a t u r e s (2) F i n i t e e l e m e n t w i t h n o d a l t e m p e r a t u r e s (3) Ana 1 y t i c a l - p 1 a n e s t r a i n (4) A n a l y t i c a l - a x i s y m r a e t r i c . 101 to t h e I n c o m p a t i b i l i t y i n t h e a p p r o x i m a t i o n o f i n i t i a l s t r a i n i n c o m p a r i s o n w i t h the s t r a i n f i e l d i n t h e l i n e a r e l e m e n t . I f t h e n o d a l t e m p e r a t u r e s a r e us e d , w i t h o u t a v e r a g i n g , b o t h t h e t e m p e r a t u r e and i n i t i a l s t r a i n v a r y l i n e a r l y w i t h i n t h e e l e m e n t w h i l e t h e l i n e a r d i s p l a c e m e n t f o r m u l a t i o n g i v e s a c o n s t a n t s t r a i n w i t h i n t h e e l e m e n t , w h i c h i s i n c o n s i s t e n t . When t h e n o d a l t e m p e r a t u r e s a r e a v e r a g e d and assumed c o n s t a n t w i t h i n the e l e m e n t , t h e i n i t i a l s t r a i n i s a l s o c o n s t a n t and i s c o n s i s t e n t w i t h t h e s t r a i n a s s o c i a t e d w i t h t h e d i s p l a c e m e n t f i e l d . The a z i m u t h a l and a x i a l s t r e s s e s , as a f u n c t i o n o f r / r Q f o r the same s o l u t i o n a r e shown i n F i g u r e s 8.13 and 8.14 r e s p e c t i v e l y . The r e s u l t s i n F i g u r e 8.13 a r e g e n e r a l l y s i m i l a r t o the r a d i a l s t r e s s e s i n F i g u r e 8.10 w i t h t h e f i n i t e e l e m e n t n o d a l t e m p e r a t u r e method of i n t r o d u c i n g t h e r m a l s t r a i n g i v i n g a c u r v e f a l l i n g w e l l below the o t h e r t h r e e . The p l a n e s t r a i n a n a l y t i c a l s o l u t i o n d e v i a t e s from t h e o t h e r two a s m a l l amount w i t h i n c r e a s i n g r / r ^ . The a x i a l s t r e s s c u r v e s i n F i g u r e 8.14 have a d i f f e r e n t c o n f i g u r a t i o n t h a n t h e r a d i a l and a z i m u t h a l s o l u t i o n s . The b e s t agreement i s between t h e f i n i t e e l ement a v e r a g e t e m p e r a t u r e c u r v e and t h e a x i s y m m e t r i c c u r v e as b e f o r e . The f i n i t e e l e m e n t n o d a l t e m p e r a t u r e c u r v e i s not v a l i d f o r t h e r e a s o n s g i v e n p r e v i o u s l y . The a n a l y t i c a l p l a n e s t r a i n v a l u e s d e v i a t e s from the o t h e r two g i v i n g maximum t e n s i l e and c o m p r e s s i v e s t r e s s e s . To t e s t t h e c o n v e r g e n c e o f t h e f i n i t e e l ement p r o c e d u r e , s t r e s s c a l c u l a t i o n s were made w i t h r e f i n e d meshes f o r a r a d i a l F i g u r e 8.15 Four s t e p s i n the mesh r e f i n e m e n t used t o a n a l y s e t h e c o n v e r gency of the f i n i t e e l ement s t r e s s c a l c u l a t i o n s . (a) NN = 15 NE = 16 (b) NN = 45, NE = 16 ( c ) NN = 45, NE = 64. (d) NN 153, NE = 256 (b) Q u a d r a t i c e l e m e n t s . ( a ) , ( c ) and (d l i n e a r e l e m e n t s . ,_, o to 103 0 - 0 2 - 0 3 - 0 - 4 CM I •o. - 0 - 5 2 - 0 - 6 - 0 7 - 0 - 8 - 0 - 9 ' NODES ELEMENTS L 45 64 • 1 Q 45 16 A L 153 256 L 15 16 • 2 L 45 64 A Q 45 16 O L 153 256 10 F i g u r e 8.16 C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n of r / r f o r d i f f e r e n t numbers of nodes and s i z e s of element (1) F i n i t e element with averaged element temperature (2) F i n i t e element with nodal temperatures. L = Lin e a r element, Q = Quadratic element. 104 t e m p e r a t u r e f i e l d . C a l c u l a t i o n s were made as a f u n c t i o n of r / r ^ f o r a c y l i n d e r o f h a l f a r a d i u s i n l e n g t h . The d i f f e r e n t s t e p s i n the mesh r e f i n e m e n t a r e shown i n F i g u r e 8.15. The r e s u l t s o f the s t r e s s c a l c u l a t i o n s a r e shown i n F i g u r e 8.16 u s i n g i n d i v i d u a l node t e m p e r a t u r e s and a v e r a g e e l e m e n t t e m p e r a t u r e s . The r e s u l t s i n d i c a t e t h a t l a r g e changes i n the number o f nodes and e l e m e n t s have no s i g n i f i c a n t e f f e c t on t h e c a l c u l a t e d s t r e s s e s i n b o t h c a s e s . The r e s u l t s a l s o show t h a t the l a r g e d i f f e r e n c e between th e n o d a l and a v e r a g e d e l e m e n t t e m p e r a t u r e i s not a s s o c i a t e d w i t h e l e m e n t s i z e but w i t h t h e i n c o m p a t i b i l i t y between s t r a i n s as d e s c r i b e d above. I t i s a l s o e v i d e n t t h a t t h e q u a d r a t i c e l e m e n t f o r m u l a t i o n does not l e a d t o a s i g n i f i c a n t d i f f e r e n c e i n the computed s t r e s s e s . F o r t h e same number of nodes fewer and l a r g e r e l e m e n t s a r e i n v o l v e d i n t h e q u a d r a t i c e l e m e n t f o r m u l a t i o n , as seen i n F i g u r e 8.16, and hence th e improved a c c u r a c y o f t h e q u a d r a t i c i n t e r p o l a t i o n of d i s p l a c e m e n t i s compromised by t h e c o a r s e r g r i d . The f i n i t e e l e m e n t and a n a l y t i c a l r a d i a l s t r e s s f i e l d s f o r an a x i s y m m e t r i c t h e r m a l f i e l d a r e shown i n F i g u r e 8.17. The t e m p e r a t u r e f i e l d c o r r e s p o n d s t o an h v a l u e o f 0.3 cm * f o r a c r y s t a l of r a d i u s 20 mm and l e n g t h 40 mm. The number o f nodes and e l e m e n t s were 256 and 153 r e s p e c t i v e l y . The f i n i t e e l e m e n t a v e r a g e t e m p e r a t u r e c u r v e n e a r l y c o i n c i d e s w i t h t h e a n a l y t i c a l a x i s y m m e t r i c c u r v e f o r r / r Q above 0.5, d e v i a t i n g below t h i s v a l u e . As b e f o r e , t h e f i n i t e e l e m e n t n o d a l t e m p e r a t u r e c u r v e d i f f e r s m a r k e d l y from t h e o t h e r s . The a n a l y t i c a l p l a n e s t r a i n c u r v e has a h i g h e r c o m p r e s s i v e s t r e s s t h a n the o t h e r s . 105 F i g u r e 8.17 C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n of r / r f o r a x i s y m m e t r i c t h e r m a l f i e l d s (1) F i n i t e e l e m e n t w i t h a v e r a g e d e l e m e n t t e m p e r a t u r e s (2) F i n i t e e l ement w i t h n o d a l t e m p e r a t u r e s (3) A n a l y t i c a l - p l a n e s t r a i n (4) A n a l y t i c a l - a x i s y m m e t r i c . 106 The d i f f e r e n c e between the a n a l y t i c a l a x i s y m m e t r i c c u r v e and f i n i t e e l e m e n t a v e r a g e t e m p e r a t u r e c u r v e i s a t t r i b u t e d t o the s l o w c o n v e r g e n c e o f t h e F o u r i e r s e r i e s combined w i t h t h e use o f t h e f i r s t o r d e r B e s s e l f u n c t i o n i n the a n a l y t i c a l s o l u t i o n of the s t r e s s f i e l d w h i c h r e s u l t s i n l e s s p r e c i s e v a l u e s . C o n v e r g e n c e i n t h i s s e r i e s i s o s c i l l a t o r y and v a l u e s were c a l c u l a t e d by a v e r a g i n g the l a s t two terms i n summations i n v o l v i n g about 40 t e r m s . The argument i n c r e a s e s l i n e a r l y w i t h the number of terms and t h e 1^ B e s s e l f u n c t i o n s d i v e r g e v e r y r a p i d l y w i t h t h e s e a r g u m e n t s . T h i s e f f e c t i s n o t s e r i o u s when r a d i a l t e m p e r a t u r e f i e l d s a r e used b e c a u s e of t h e s i m p l i c i t y of t h e f u n c t i o n w hich i s expanded ( t h e u n i t f u n c t i o n ) , as opposed to a complex z - d e p e n d e n t f u n c t i o n w hich i s expanded i n t h e c a s e of an a x i s y m m e t r i c two d i m e n s i o n a l t e m p e r a t u r e f i e l d . The i n c l u s i o n of t h e I f u n c t i o n s was done i n o r d e r t o o b t a i n a more g e n e r a l s o l u t i o n t o t h e i s o t h e r m a l d i s p l a c e m e n t d i f f e r e n t i a l e q u a t i o n . R e c e n t l y s o l u t i o n s f o r s i m i l a r e q u a t i o n s have been o b t a i n e d i n s e m i - i n f i n i t e c y l i n d e r s i n c l u d i n g o n l y the J B e s s e l f u n c t i o n s i n 16 8 a l e s s g e n e r a l but b e t t e r behaved s o l u t i o n The r e s u l t s i n F i g u r e 8.17 a l s o show t h a t f o r a x i s y m m e t r i c t e m p e r a t u r e f i e l d s t h e p l a n e s t r a i n a p p r o x i m a t i o n g i v e s s t r e s s e s w h i c h a r e more t h a n d o u b l e the s t r e s s e s o b t a i n e d w i t h t h e f i n i t e e l e m e n t method u s i n g t h e a x i s y m m e t r i c a p p r o x i m a t i o n s . L a r g e r d i f f e r e n c e s were u s u a l l y o b s e r v e d c l o s e to the ends o f l o n g e r c r y s t a l where c o n d i t i o n s d e v i a t e m a r k e d l y from p l a n e s t r a i n . T h i s r e s u l t shows t h a t t h e p l a n e s t r a i n a p p r o x i m a t i o n i s not always good f o r t h e a n a l y s i s o f t h e r m a l l y i n d u c e d s t r e s s e s . I t s h o u l d be 107 n o t e d t h a t t h i s r e s u l t has not been r e p o r t e d i n t h e l i t e r a t u r e 2 1 4 " 2 1 7 . In summary, i t has been shown t h a t t h e r e i s good agreement between the f i n i t e e l e m e n t a n a l y s i s u s i n g a v e r a g e e l e m e n t t e m p e r a t u r e s and t h e a n a l y t i c a l s o l u t i o n w i t h t h e a x i s y m m e t r i c a s s u m p t i o n . The a n a l y t i c a l s o l u t i o n w i t h the p l a n e s t r a i n a s s u m p t i o n does not f i t as w e l l due t o t h e r e s t r i c t i o n o f a x i a l movement i n t h i s c a s e . I t may t h u s be c o n c l u d e d t h a t the l i n e a r f i n i t e e l e m e n t a n a l y s i s g i v e s s t r e s s e s t h a t a r e c o m p a r a b l e t o t h e a x i s y m m e t r i c a p p r o x i m a t i o n and w i t h the p l a n e s t r a i n a p p r o x i m a t i o n when t h a t c o n d i t i o n a p p l i e s . 8.3 C o m p a r i s o n o f Model P r e d i c t i o n s w i t h T e m p e r a t u r e Measurements The t e m p e r a t u r e d i s t r i b u t i o n i n a g r o w i n g c r y s t a l c a l c u l a t e d from the model i s compared to the r e s u l t s of t e m p e r a t u r e 218 measurements made i n a LEC GaAs M e l b o u r n c r y s t a l g r o wer. The t e m p e r a t u r e measurements were made w i t h t h e a r r a y of f o u r t h e r m o c o u p l e s shown i n F i g u r e 8.18. The t h e r m o c o u p l e s were a t t a c h e d to the seed h o l d e r and moved up w i t h the s e e d as t h e c r y s t a l was grown. T e m p e r a t u r e s were r e c o r d e d f o r t h e e n t i r e g rowth p e r i o d of the c r y s t a l , g i v i n g the r e s u l t s shown i n F i g u r e 8.19. To c a l c u l a t e t h e t e m p e r a t u r e d i s t r i b u t i o n w i t h t h e model, 2 the t h e r m a l d i f f u s i v i t y was t a k e n as 0.04 cm / s . The h e a t t r a n s f e r c o e f f i c i e n t s between th e c r y s t a l and the a r g o n and B O were d e t e r m i n e d from the v a l u e s g i v e n i n s e c t i o n 8.1. 108 F i g u r e 8.18 P o s i t i o n o f t h e r m o c o u p l e s i n GaAs c r y s t a l . B = B o r i c o x i d e l a y e r , a r g o n p r e s s u r e 3.04 MPa. R e f e r e n c e 218. 109 The ambient t e m p e r a t u r e d i s t r i b u t i o n a l o n g t h e c r y s t a l , shown i n F i g u r e 8.19 ( c u r v e 4), i s b a s e d on t e m p e r a t u r e measurements a d j a c e n t to a f i x e d p o i n t on t h e c r y s t a l a t d i f f e r e n t t i m e s d u r i n g growth, c o n v e r t i n g t i m e t o d i s t a n c e w i t h the growth v e l o c i t y . I t i s assumed t h a t t h i s t e m p e r a t u r e d i s t r i b u t i o n i s t i m e i n d e p e n d e n t . On the b a s i s o f t h i s a s s u m p t i o n , t h e measured ambient t e m p e r a t u r e i s u s e d t o d e t e r m i n e t h e h e a t t r a n s f e r c o e f f i c i e n t at each segment a l o n g the s u r f a c e o f t h e g r o w i n g c r y s t a l . The t e m p e r a t u r e d i s t r i b u t i o n w i t h i n t h e c r y s t a l w i l l change w i t h time as the c r y s t a l i s p u l l e d from t h e m e l t . The c a l c u l a t e d and measured t e m p e r a t u r e s a l o n g t h e c r y s t a l a x i s a t f o u r d i f f e r e n t c r y s t a l l e n g t h s a r e shown i n F i g u r e 8.20. A c o m p a r i s o n o f t h e c a l c u l a t e d and measured t e m p e r a t u r e s i s o n l y v a l i d a t the p o i n t where the c a l c u l a t e d t e m p e r a t u r e c o i n c i d e s w i t h t h e measured t e m p e r a t u r e , a t p o i n t s marked A, B, C and D i n F i g u r e 8.20 f o r t h e f o u r c r y s t a l l e n g t h s . Agreement between the c a l c u l a t e d and measured t e m p e r a t u r e s at t h e s e p o i n t s i s c l o s e ; w i t h i n 15°C. F o r t h e s h o r t c r y s t a l (55.0 mm), t h e measured and c a l c u l a t e d c u r v e s e f f e c t i v e l y c o i n c i d e , i n d i c a t i n g t h a t the measured c u r v e i s t i m e i n d e p e n d e n t . F o r the l o n g e r c r y s t a l s the measured c u r v e s a r e below th e c a l c u l a t e d c u r v e i n d i c a t i n g t e m p e r a t u r e d e pendence, i . e . t h a t t h e t e m p e r a t u r e a t a g i v e n p o i n t d e c r e a s e s a f t e r b e i n g measured by the t h e r m o c o u p l e as the c r y s t a l grows. The d i f f e r e n c e i n c r e a s e s w i t h i n c r e a s i n g c r y s t a l l e n g t h . 110 •20 0 20 4 0 60 80 100 120 140 160 Position Relative To Interface (mm) F i g u r e 8.19 T e m p e r a t u r e s measured w i t h t h e r m o c o u p l e s 2, 3 and 4 i n F i g u r e 8.18 as a f u n c t i o n o f t h e r e l a t i v e p o s i t i o n o f t h e t h e r m o c o u p l e s w i t h t h e i n t e r f a c e . R e f e r e n c e 218. I l l — — i 1 1 1 1 1 1 1 1 1 r 1300 Position Relative To Interface (mm) F i g u r e 8.20 Measured and c a l c u l a t e d temperatures c r y s t a l a x i s at four c r y s t a l l e n g t h s . along the 112 Fi g u r e 8.21 Measured and c a l c u l a t e d te ou t s i d e s u r f a c e of the l e n g t h s . The measured ambi shown. mperatures adjacent to the c r y s t a l at four c r y s t a l ent temperature i s a l s o 113 81 1 1 1 1 1 1 1 1 1 1 r l 1 1 1 1 1 i i • • I L 0 20 4 0 60 80 100 Position Relative To Interface (mm) F i g u r e 8.22 Measured and c a l c u l a t e d a x i a l temperature g r a d i e n t s along the c r y s t a l a x i s at four c r y s t a l l e n g t h s . 114 The t e m p e r a t u r e d i s t r i b u t i o n n e a r the edge o f the c r y s t a l , b o t h c a l c u l a t e d and measured, f o r f o u r c r y s t a l l e n g t h s i s shown i n F i g u r e 8.21. The ambient t e m p e r a t u r e d i s t r i b u t i o n i s a l s o i n c l u d e d i n t h e f i g u r e . Note t h a t a d j a c e n t t o t h e i n t e r f a c e , c u r v e (a) was measured and c u r v e (b) was u s e d as the ambient t e m p e r a t u r e i n the model s i n c e t h e i n t e r f a c e i s assumed f l a t and at t h e f r e e z i n g t e m p e r a t u r e . C o mparing t h e c a l c u l a t e d and measured v a l u e s a t p o i n t s A, B, C and D shows t h e r e i s r e a s o n a b l e a greement between the two v a l u e s . The c a l c u l a t e d v a l u e s a r e l o w e r t h a n t h e measured v a l u e s by l e s s t h a n 25°C. As w i t h t h e a x i a l t e m p e r a t u r e s the measured v a l u e s a r e below the c a l c u l a t e d v a l u e s o v e r most o f the c r y s t a l , t h e d i f f e r e n c e i n c r e a s i n g w i t h i n c r e a s i n g c r y s t a l l e n g t h . The a x i a l g r a d i e n t was a l s o measured d u r i n g growth by t h e r m o c o u p l e s 1 and 2 i n F i g u r e 8.18. The r e s u l t s a r e compared t o the c a l c u l a t e d g r a d i e n t s i n F i g u r e 8.22. The measured g r a d i e n t s a r e s i g n i f i c a n t l y below the c a l c u l a t e d g r a d i e n t s near t h e i n t e r f a c e . At t h e s e p o i n t s the measured g r a d i e n t s may c o r r e s p o n d p a r t l y t o t h e melt and p a r t l y t o t h e c r y s t a l . In a d d i t i o n , t h e p r e s e n c e o f a convex i n t e r f a c e , as i s shown l a t e r i n t h i s work, g i v e s l o w e r a x i a l g r a d i e n t s t h a n a f l a t i n t e r f a c e i n t h i s r e g i o n . F o r l o n g e r c r y s t a l s the measured g r a d i e n t s a r e about 10 % below t h e c a l c u l a t e d v a l u e s , t h e d i f f e r e n c e i n c r e a s i n g w i t h c r y s t a l l e n g t h . At the p o i n t s where the c o m p a r i s o n i s v a l i d the measured v a l u e s a r e above the c a l c u l a t e d v a l u e s but w i t h i n t h e range of the s c a t t e r of e x p e r i m e n t a l measurements. 115 In summary, the v a l i d p o i n t s of c o m p a r i s o n o f t h e c a l c u l a t e d and measured t e m p e r a t u r e s a r e i n good a g r e e m e n t . A c c o r d i n g l y , the t e m p e r a t u r e s p r e d i c t e d by the model a r e i n d i c a t i v e o f the r e a l t e m p e r a t u r e s i n t h e c r y s t a l . 8.4 The T e m p e r a t u r e Model f o r C o o l i n g - Programming and C o n v e r g e n c y To c a l c u l a t e t h e t e m p e r a t u r e f i e l d s b a s e d on t h e a n a l y t i c a l t i m e d e p e n d e n t s o l u t i o n s , a FORTRAN program was w r i t t e n . T h i s program i s shown i n A p p e n d i x X I . The f l o w c h a r t o f t h i s program i s shown i n F i g u r e 8.23. The v a l u e of X were c a l c u l a t e d u s i n g 219 c u b i c i n t e r p o l a t i o n f r o m t a b l e s , t h e f i r s t s i x r o o t s were u s e d . The Y - v a l u e s were o b t a i n e d from E q u a t i o n (7.41b) u s i n g a Newton - Raphson method t o f i n d t h e r o o t s . The f i r s t 30 v a l u e s were u s e d . W i t h t h e s e s i x X v a l u e s , the r a d i a l p a r t of the t e m p e r a t u r e c o n v e r g e d t o 98 % o f i t s f i n a l v a l u e . W i t h 30 Y v a l u e s the a x i a l p a r t o f t h e t e m p e r a t u r e c o n v e r g e d t o 98 % of i t s f i n a l v a l u e . The n o n - d i m e n s i o n a l t e m p e r a t u r e was t h e r e f o r e c a l c u l a t e d as 96 * of the e x a c t s o l u t i o n . A t y p i c a l t e m p e r a t u r e f i e l d c a l c u l a t e d d u r i n g c o o l i n g i s shown i n F i g u r e 8.24 a. The r a d i u s o f t h e c r y s t a l i s 27.5 mm and the l e n g t h i s 110.0 mm. Due t o symmetry, o n l y t h e r i g h t h a l f of the c r y s t a l i s shown. C l o s e to t h e i n t e r f a c e or t a i l end the a x i a l t e m p e r a t u r e g r a d i e n t changes from p o s i t i v e t o n e g a t i v e . A l s o a h i g h r a d i a l g r a d i e n t i s u s u a l l y o b s e r v e d as opposed to what i s o b s e r v e d d u r i n g g rowth. In v i e w o f t h i s and t h e f a c t t h a t 116 ( START / I N P U T - y eig e n v a l u e s - 3 e i g e n v a l u e s - c r y s t a l r a d i u s and length - heat t r a n s f e r c o e f f i c i e n t , h - ambient temperature - i n i t i a l a x i a l temperature For each 3 : CALCULATE c o e f f i c i e n t (3) For each Y : CALCULATE c o e f f i c i e n t C (y) For each r a d i a l p o s i t i o n : EVALUATE s e r i e s i n Bessel f u n c t i o n s 5 117 1 F o r e a c h a x i a l p o s i t i o n EVALUATE s e r i e s i n F o u r i e r f u n c t i o n s F o r e a c h p o i n t i n the g r i d CALCULATE t e m p e r a t u r e / / / PRINT n o d a l t e m p e r a t u r e s / j PLOT i s o t h e r m s / STOP F i g u r e 8.23 Flow c h a r t o f t h e computer program f o r t h e n u m e r i c a l e v a l u a t i o n of t h e a n a l y t i c a l t e m p e r a t u r e f i e l d s d u r i n g c o o l i n g o f t h e c r y s t a l . 118 a b e F i g u r e 8.24 (a) T y p i c a l temperature f i e l d o b t a i n e d d u r i n g c o o l i n g u n i t s a r e 10 °C. (b) and ( c ) Von M i s e s s t r e s s c o n t o u r s (MPa) f o r the t e m p e r a t u r e f i e l d g i v e n i n ( a ) . (b) NN = 451, NE = 800. ( c ) NN = 1105, NE = 2048. 119 f o r the s t r e s s c a l c u l a t i o n a v e r a g e t e m p e r a t u r e s a r e us e d a n o t h e r mesh r e f i n e m e n t was p e r f o r m e d . T h i s was done t o d e t e r m i n e i f f o r a g i v e n mesh s i z e the c a l c u l a t e d s t r e s s e s were r e p r e s e n t a t i v e . The r e s u l t s are shown i n F i g u r e 8.24 b and c. F i g u r e 8.24 b shows the Von M i s e s s t r e s s f i e l d I n d u c e d by the t e m p e r a t u r e f i e l d i n p a r t (a) of the f i g u r e , c a l c u l a t e d u s i n g a mesh c o n t a i n i n g 451 nodes and 800 e l e m e n t s . F i g u r e 8.24 c shows t h e same s t r e s s f i e l d o b t a i n e d e m p l o y i n g a mesh o f 1105 nodes and 2048 e l e m e n t s . I t i s s e e n t h a t e x a c t l y the same s t r e s s p a t t e r n i s o b t a i n e d w i t h s t r e s s v a l u e s which a r e about 10 * l e s s i n the c o a r s e mesh f o l l o w i n g an e x p e c t e d t e n d e n c y . The c o m p u t i n g e f f o r t i n v o l v e d , however i n c r e a s e d q u a d r a t i c a l l y w i t h t h e number of nodes. A c c o r d i n g l y the c o a r s e r mesh was s e l e c t e d . 120 CHAPTER 9 RESULTS AND ANALYSIS The m a t h e m a t i c a l model p r e s e n t e d i n t h e p r e v i o u s c h a p t e r s i s used t o s t u d y t h e e f f e c t of the d i f f e r e n t v a r i a b l e s i n g e n e r a t i n g d i s l o c a t i o n s d u r i n g growth and c o o l i n g to ambient t e m p e r a t u r e . The v a r i a b l e s r e l a t e d to c r y s t a l g eometry d u r i n g growth i n c l u d e t h e f o l l o w i n g 1. Cone a n g l e (CA), g i v e n by t h e a n g l e between the cone s u r f a c e and the h o r i z o n t a l 2. C r y s t a l l e n g t h ( C L ) , g i v e n by t h e d i s t a n c e between the s o l i d - l i q u i d i n t e r f a c e and t h e s t a r t of the cone 3. C r y s t a l r a d i u s ( R ) , 4. I n t e r f a c e shape The v a r i a b l e s a s s o c i a t e d w i t h t h e g rowth p r o c e s s i n c l u d e t h e f o l l o w i n g : 5. Growth v e l o c i t y (V) 6. Boron o x i d e t h i c k n e s s (B) 7. T h e r m a l g r a d i e n t s ( a r g o n , AG and b o r o n o x i d e , BG) 8 . Gas p r e s s u r e (GP) 9. Gas c o m p o s i t i o n The v a r i a b l e s a s s o c i a t e d w i t h t h e c o o l i n g p r o c e s s i n c l u d e the f o l l o w i n g : 121 10. The n a t u r e of t h e media i n wh i c h t h e c r y s t a l i s immersed. T h i s may be bor o n o x i d e or a r g o n gas 11. T e m p e r a t u r e o f the c o o l i n g media 12. Thermal c o n d i t i o n s i n the c r y s t a l a t t h e s t a r t of c o o l i n g . The e f f e c t of t h e above v a r i a b l e s on t h e c a l c u l a t e d s t r e s s f i e l d s and d i s l o c a t i o n d e n s i t y a r e now c o n s i d e r e d . 9.1 Cone A n g l e C r y s t a l and Growth P a r a m e t e r s Assumed R, 20 mm ; CL, 10 mm ; B, 10 mm ; BG , 100 °C/cm ; AP, 30 atm ; AG, 50°C/cm. The cone a n g l e s c o n s i d e r e d a r e 7.1, 30, 45, 54.7 and 65 d e g r e e s . The a n g l e 5 4 . 7 ° c o r r e s p o n d s t o t h e cone s u r f a c e c o i n c i d e n t w i t h t h e (111) p l a n e . The c a l c u l a t e d Von M i s e s s t r e s s d i s t r i b u t i o n on a v e r t i c a l p l a n e f o r the f i v e cone a n g l e s c o n s i d e r e d i s shown i n F i g u r e 9 . 1 ( a - e ) . The maximum Von M i s e s S t r e s s (MVMS) l e v e l s o c c u r at the s h o u l d e r where t h e cone r e a c h e s the f u l l r a d i u s o f the c r y s t a l . H i g h s t r e s s l e v e l s o c c u r a l o n g b o t h t h e cone s u r f a c e and t h e c r y s t a l a x i s . The l o w e s t Von M i s e s s t r e s s e s (LVMS) a r e below the se e d . 122 123 F i g u r e 9.1 Von M i s e s S t r e s s c o n t o u r s (MPa) i n v e r t i c a l p l a n e s f o r f i v e cone a n g l e s . (a) 7 . 1 ° , (b) 3 0 ° , ( c ) 45 , (d) 5 4 . 7 ° , (e) 6 5 ° . Cone s u r f a c e i n (d) c o i n c i d e s w i t h a (111) p l a n e . C r y s t a l r a d i u s , 20 mm ; c r y s t a l l e n g t h , 10 mm ; B 2 ° 3 t h i c k n e s s , 10 mm ; B 2 ° 3 g r a d i e n t , 100 C/cm ; a r g o n p r e s s u r e , 30 atm. ; a r g o n g r a d i e n t , 50 C/cm. 124 \Table 9.1 E f f e c t of Cone A n g l e on Th e r m a l and S t r e s s F i e l d s T h e r m a l G r a d i e n t s , °C/cm| 1 % V™ rOue* 4 s l _ : MVMS I AMRSS 1 I AMRSS/ Cone A n g l e ( ° ) | R a d i a l A x i a l | MPa MPa MVMS 55 18.4 8 . 7 0 .47 50 6 . 7 3 . 1 0 . 46 54 5 . 2 2 . 5 0 . 48 56 1 . 1 2 . 6 2 . 4 55 5 . 0 2 . 3 0 . 45 7.1 6.7 30 . 4.1 45. 2.7 54.7 3.2 65 . 2.5 The l a r g e s t MVMS, as shown i n T a b l e 9.1, o c c u r s i n the c r y s t a l w i t h the s h a r p e s t cone a n g l e of 7 . 1 ° . The MVMS d r o p s a p p r e c i a b l y when the cone a n g l e i s i n c r e a s e d from 7.1° to 3 0 ° , and d e c r e a s e s f u r t h e r by a s m a l l e r amount between 30° and 45° . The MVMS r e a c h e s a minimum f o r a cone a n g l e of 57.4° t h e n i n c r e a s e s between 52.4° and 65° . S i m i l a r b e h a v i o u r i s o b s e r v e d when t h e LVMS i s c o n s i d e r e d as a f u n c t i o n of cone a n g l e . In o r d e r t o a c c o u n t f o r t h e e f f e c t of t h e cone a n g l e on the VMS two s o u r c e s of t h e r m a l s t r e s s a r e c o n s i d e r e d namely the t e m p e r a t u r e f i e l d s and t h e g e o m e t r i c a l c o n s t r a i n t s . The t e m p e r a t u r e f i e l d i s the o r i g i n o f t h e r m a l s t r e s s e s . The s t r e s s l e v e l i s r o u g h l y d e t e r m i n e d by t h e t h e r m a l g r a d i e n t s . G e n e r a l l y , l a r g e r a d i a l t h e r m a l g r a d i e n t s combined w i t h l a r g e a x i a l g r a d i e n t s p r o d u c e l a r g e s t r e s s e s i n c y l i n d e r s . I f t h e a x i s y m m e t r i c body does not have c o n s t a n t c r o s s - s e c t i o n , a d d i t i o n a l s t r e s s e s may d e v e l o p due t o g e o m e t r i c a l c o n s t r a i n t s . 125 F o r the c r y s t a l g e o m e t r y c o n s i d e r e d h e r e t e m p e r a t u r e g r a d i e n t s a r e c a l c u l a t e d as f o l l o w s . The a x i a l g r a d i e n t s a r e c a l c u l a t e d from the t e m p e r a t u r e d i f f e r e n c e s between t h e i n t e r f a c e and a p o i n t i n t h e c r y s t a l a x i s 10 mm above, l a b e l l e d A and B r e s p e c t i v e l y i n F i g u r e 9.1(e) . The r a d i a l g r a d i e n t s a r e c a l c u l a t e d from t h e t e m p e r a t u r e d i f f e r e n c e s between p o i n t B and C which a r e a l s o shown i n F i g u r e 9 . 1 ( e ) . The r e s u l t s a r e g i v e n i n T a b l e 9.1. From T a b l e 9.1 t h e a x i a l g r a d i e n t s a r e o b s e r v e d t o be r e l a t i v e l y i n s e n s i t i v e t o t h e cone a n g l e , v a r y i n g by abo u t 3 * about t h e mean g r a d i e n t o f 53°c/cm. On t h e o t h e r hand t h e r a d i a l g r a d i e n t s v a r y a p p r e c i a b l y about t h e mean g r a d i e n t o f 4.2°C/cm. The g r a d i e n t d r o p s s h a r p l y from 6.7°C/cm between 7.1° and 3 0 ° , dr o p s f u r t h e r a t 4 5 ° , r i s e s a t 54.7 ° and t h e n d r o p s a g a i n a t 6 5 ° . C o n s i d e r i n g the MVMS v a l u e s i n T a b l e 9.1, t h e drop i n MVMS between 7.1° and 30° can be a c c o u n t e d f o r by t h e change i n r a d i a l g r a d i e n t . The minimum MVMS f o r 54.7° i s a t t r i b u t e d t o t h e g e o m e t r i c a l c o n s t r a i n t s a r i s i n g from the v a r i a t i o n s i n r a d i u s of the c r y s t a l a t t h e cone, as w e l l as t h e r a d i a l g r a d i e n t . F o r a z e r o cone a n g l e , c o n d i t i o n s s i m i l a r t o p l a i n s t r e s s w i l l p r e v a i l a t t h e cone s u r f a c e , w i t h z e r o a x i a l and s h e a r components o f t h e a x i s y m m e t r i c s t r e s s t e n s o r . Under s u c h c o n d i t i o n s the VMS w i l l have maximum v a l u e s b e c a u s e t h e r e a r e o n l y two l a r g e n o n - z e r o d i a g o n a l s t r e s s components. F o r t h e same t h e r m a l c o n d i t i o n s , when the cone i s not z e r o a s i m i l a r s t r e s s l e v e l w i l l have a d i f f e r e n t d i s t r i b u t i o n of s t r e s s components g i v i n g a t h i r d n o n - z e r o 126 p r i n c i p a l s t r e s s . The d i f f e r e n c e between t h e p r i n c i p a l s t r e s s components d e c r e a s e s as t h e cone a n g l e i n c r e a s e s , g i v i n g t h e o b s e r v e d VMS dependence w i t h cone a n g l e . F o r cone a n g l e s l a r g e r t h a n 5 4 . 7 ° a n o t h e r f a c t o r s h o u l d be i n c l u d e d i n the a n a l y s i s . Under t h e s e c o n d i t i o n s the c r y s t a l s a r e e f f e c t i v e l y l a r g e r t h a n t h e low cone a n g l e c r y s t a l s . T h e r e f o r e t h e c o n s t r a i n t s a r e such t h a t c o n d i t i o n s c l o s e r to t h o s e of p l a n e s t r a i n a p p l y . In such c a s e s i t was shown i n t h e l a s t c h a p t e r t h a t s t r e s s e s a r e l a r g e r t h a n under the more g e n e r a l a x i s y m m e t r i c c o n d i t i o n s . From the above d i s c u s s i o n , a t low cone a n g l e s t h e VMS i s e x p e c t e d to d e c r e a s e w i t h i n c r e a s i n g cone a n g l e . At h i g h cone a n g l e s , however, the VMS i s e x p e c t e d t o i n c r e a s e w i t h i n c r e a s i n g cone a n g l e . T h e r e f o r e a minimum i n the VMS s h o u l d e x i s t a t i n t e r m e d i a t e cone a n g l e s . In t h i s c a s e t h a t minimum i s o b s e r v e d f o r a cone a n g l e of 5 4 . 7 ° . The maximum component of t h e r e s o l v e d s h e a r s t r e s s (MRSS) i n v e r t i c a l (010) p l a n e s f o r t h e f i v e cone a n g l e s c o n s i d e r e d i s shown i n F i g u r e 9 . 2 ( a - e ) . The MRSS s t r e s s d i s t r i b u t i o n s a r e s i m i l a r to the VMS d i s t r i b u t i o n s . The dependence of the MRSS on the cone a n g l e i s d e t e r m i n e d by c o n s i d e r i n g t h e a b s o l u t e maximum MRSS (AMRSS) d e v e l o p e d i n t h e c r y s t a l which i s r e p r e s e n t a t i v e of th e s t r e s s l e v e l i n the whole c r y s t a l . The AMRSS o c c u r s at the s h o u l d e r as i s shown i n the VMS d i s t r i b u t i o n s . The AMRSS f o r the a n g l e s c o n s i d e r e d i s l i s t e d i n T a b l e 9.1 column 5. F o r low cone a n g l e s the AMRSS d e c r e a s e s r a p i d l y s i m i l a r t o t h a t shown f o r the VMS. F o r cone a n g l e s of 4 5 ° and l a r g e r t h e AMRSS r e m a i n s 127 128 F i g u r e 9.2 Maximum r e s o l v e d s h e a r s t r e s s (MRSS) c o n t o u r s i n MPa, i n v e r t i c a l (010) p l a n e s f o r f i v e cone a n g l e s , (a) 7.1°, (b) 3 0 ° , ( c ) 4 5 ° , (d) 5 4 . 7 ° , (e) 6 5 ° . Cone s u r f a c e i n (d) c o i n c i d e s w i t h a (111) p l a n e . C o n d i t i o n s a r e t h e same as i n F i g u r e 9.1 . 129 e s s e n t i a l l y c o n s t a n t . The l a r g e d r o p i n t h e AMRSS a t low cone a n g l e s s u p p o r t s the c o n c l u s i o n t h a t t h e s t r e s s e s a r e dep e n d e n t e n t i r e l y on t h e t h e r m a l f i e l d c h a n g e s r a t h e r t h a n on g e o m e t r i c a l c o n s t r a i n t s . At l a r g e cone a n g l e s , t h e MRSS d i f f e r s from t h e VMS. The d i f f e r e n c e can be seen i n T a b l e 9.1 column 6 where the AMRSS/MVMS i s l i s t e d . F o r a l l cone a n g l e s e x c e p t 5 4 . 7 ° t h e AMRSS i s l e s s t h a n h a l f t h e MVMS. At 54.7° a s i n g u l a r i t y o c c u r s s u c h t h a t t h e AMRSS i s more t h a n d o u b l e t h e MVMS. The s i n g u l a r i t y a t 5 4 : 7 ° cone a n g l e i s a t t r i b u t e d t o t h e model a s s u m p t i o n of no t r a c t i o n a t the c r y s t a l s u r f a c e and the d i f f e r e n c e i n the d e f i n i t i o n of t h e VMS and RSS components. The c o n d i t i o n of no t r a c t i o n r e q u i r e s the p r o j e c t i o n o f the s t r e s s t e n s o r on a d i r e c t i o n normal t o t h e c r y s t a l s u r f a c e t o be z e r o . In t h i s c a s e , f o r a g i v e n s t r e s s l e v e l , t he s t r e s s components must accommodate i n a d i r e c t i o n p a r a l l e l t o t h e c r y s t a l s u r f a c e . F o r a cone a n g l e p r e s e n t i n g a s u r f a c e p a r a l l e l t o a (111) p l a n e , t h e s t r e s s components l i e on the (111) p l a n e . In t h i s c a s e the RSS components a r e e x p e c t e d to be a t maxima s i n c e by d e f i n i t i o n t h e y a r e p r o j e c t i o n s o n t o t h e s e p l a n e s i n t h e <110> d i r e c t i o n s . The VMS, on the o t h e r hand, i s an i n d i c a t i o n o f the s h e a r s t r e s s i n i s o t r o p i c m a t e r i a l s and t h e r e f o r e does not i n c l u d e the c r y s t a 1 1 o g r a p h i c p r o p e r t i e s of the m a t e r i a l . These r e s u l t s show the a d v a n t a g e of u s i n g the MRSS o v e r t h e VMS f o r the a n a l y s i s of s t r e s s e s g e n e r a t i n g d i s l o c a t i o n s . From the VMS r e s u l t s i t may be m i s t a k e n l y c o n c l u d e d t h a t a 5 4 . 7 ° cone a n g l e i s b e n e f i c i a l i n o r d e r to r e d u c e d i s l o c a t i o n g e n e r a t i o n . A 130 d i f f e r e n t c o n c l u s i o n can be o b t a i n e d u s i n g t h e RSS which i s t h e s t r e s s g e n e r a t i n g d i s l o c a t i o n s by g l i d e . The r e s u l t s u s i n g t h e MRSS show t h a t t h e r e i s no su c h b e n e f i c i a l e f f e c t u s i n g a 5 4 . 7 ° cone a n g l e . On t h e c o n t r a r y , i f growth c o n d i t i o n s a r e su c h t h a t s t r e s s l e v e l s g i v e n by t h e VMS a r e changed, t h e e f f e c t on d i s l o c a t i o n d e n s i t i e s i s d o u b l e a t t h i s a n g l e . The p r e s e n t c a l c u l a t i o n s d i f f e r s i g n i f i c a n t l y from t h e 13 7 c a l c u l a t i o n s r e p o r t e d by J o r d a n e t a l . . In t h e p r e s e n t c a s e the MRSS i s used, whereas J o r d a n e t a l . use the t o t a l r e s o l v e d s h e a r s t r e s s (TRSS) w h i c h i s t h e sum of t h e t w e l v e RSS v a l u e s . F o r the 45° cone a n g l e t h e TRSS d i s t r i b u t i o n shown i n F i g u r e 9.3 i s s i m i l a r to the MRSS d i s t r i b u t i o n shown i n F i g u r e 9 . 2 ( c ) , but the s t r e s s l e v e l s of the TRSS a r e s e v e n t i m e s l a r g e r t h a n t h e MRSS and t h r e e t i m e s l a r g e r t h a n the VMS l e v e l s . A d d i n g the components t o make t h e TRSS i s i n c o n s i s t e n t w i t h t h e c o n c e p t of a v e c t o r i a l or t e n s o r i a l q u a n t i t y s i n c e the magni t u d e c a n n o t be c a l c u l a t e d by a l i n e a r a d d i t i o n o f components. M o r e o v e r t h e TRSS can be i n d e f i n i t e l y i n c r e a s e d by c o n s i d e r i n g an u n l i m i t e d number of s e c o n d a r y s l i p s y s t e m s w h i c h i s a l s o i n c o n s i s t e n t . The e f f e c t i v e s t r e s s a s s o c i a t e d w i t h d i s l o c a t i o n g e n e r a t i o n and movement can be d e t e r m i n e d by s u b s t r a c t i n g from t h e MRSS t h e c r i t i c a l r e s o l v e d s h e a r s t r e s s f o r y i e l d o r t h e v a l u e r e p o r t e d f o r d i s l o c a t i o n g e n e r a t i o n . T h i s i s done i n F i g u r e 9.4 i n wh i c h (a) the y i e l d s t r e s s i s s u b s t r a c t e d from the MRSS [CRSS ( y i e l d ) ] , (b) t h e CRSS f o r d i s l o c a t i o n g e n e r a t i o n g i v e n by M i l v i d s k i i and B o c h k a r e e v 1 5 0 ' 1 6 3 i s s u b s t r a c t e d [CRSS (MB)] and ( c ) t h e CRSS 131 F i g u r e 9.3 T o t a l r e s o l v e d s h e a r s t r e s s (TRSS) c o n t o u r s i n MPa f o r t h e 45 cone a n g l e c r y s t a l . Compare t h e l a r g e s t r e s s l e v e l s o f the TRSS w i t h the s t r e s s l e v e l s f o r same c r y s t a l shown i n F i g u r e 9 . 1 ( c ) f o r the VMS and F i g u r e 9.2(c) f o r the MRSS. / 4 C o n t o u r s of the MRSS (MPa) i n e x c e s s o f : (a) CRSS ( y i e l d ) ; (b) CRSS (MB) ; ( c ) CRSS (MBTe). Shaded r e g i o n s i n d i c a t e a r e a s i n wh i c h t h e MRSS i s l e s s t h a n the CRSS. 133 f o r T e - d o p e d c r y s t a l s i s s u b s t r a c t e d [CRSS (MBTe) ] . The sha d e d a r e a s i n t h e f i g u r e s i n d i c a t e where t h e e f f e c t i v e s t r e s s i s z e r o . I t can be se e n t h a t the MRSS i s l e s s t h an t h e y i e l d s t r e s s i n most of t h e c r y s t a l e x c e p t i n t h e c e n t r e and n e a r t h e s u r f a c e . The l a r g e s t s t r e s s e s o c c u r i n a r e g i o n below the s h o u l d e r a t A. The MRSS i s a l w a y s g r e a t e r t h a n t h e CRSS (MB) as shown i n ( b ) . D o p i n g w i t h Te i n ( c ) g i v e s r e g i o n s i n t h e c r y s t a l where t h e MRSS does not e x c e e d t h e CRSS (MBTe) c o m p a r a b l e t o (a) . The s p e c i f i c d i r e c t i o n s and p l a n e s a s s o c i a t e d w i t h t h e maximum r e s o l v e d s h e a r s t r e s s e s i n t h e c r y s t a l a r e shown i n F i g u r e 9.5. P l o t s o f s t r e s s d i s t r i b u t i o n f o r o t h e r cone a n g l e s a r e g i v e n i n A p p e n d i x X I I I . These p l o t s show s i m i l a r p a t t e r n s t o t h a t shown i n F i g u r e 9.4 w i t h t h e e x c e p t i o n o f t h e c r y s t a l w i t h a cone a n g l e o f 7.1° ( F i g u r e 9 . 6 ( a - c ) ) . 9.1.1 E f f e c t of Cone A n g l e on S t r e s s Symmetry The e f f e c t o f cone a n g l e on t h e MRSS d i s t r i b u t i o n on p l a n e s p e r p e n d i c u l a r t o the growth d i r e c t i o n i s shown i n F i g u r e 9 . 7 ( a - e ) . The p l a n e s i n (a-d) a r e 7.5 mm from the i n t e r f a c e , and t h e p l a n e i n (e) i s 8 mm from the i n t e r f a c e . The v e r t i c a l d i r e c t i o n i n t h e f i g u r e s i s [010] and t h e h o r i z o n t a l d i r e c t i o n [ 1 0 0 ] . In F i g u r e 9 . 7 ( a ) , f o r the s m a l l e s t cone a n g l e ( 7 . 1 ° ) , the s t r e s s d i s t r i b u t i o n i s complex w i t h low s t r e s s e s a t t h e c e n t r e f o l l o w e d by a r i n g w i t h h i g h e r s t r e s s e s w h i c h e x h i b i t s f o u r - f o l d symmetry. 134 O-O-O-O-O-D <J> O A A A Q o o A A A LJ) 6 O A A • O o o AO o A 6 o • • o • ' \ o o o o o o. I I I I I I V I o I o o o o o x o o o o o, I o 6  V o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o\>. \ 6 o o o o o o o o o <p o o o o o o o o o o o o o oNo. o o o o o o o o o o o o o o o o o o o o o o o o o o o o o • o o. o o. ( I I I ( T i l ( TT I O ( M l 0 ( I I . ( I l l • \ o-o- o o O o o • • • 1 • c > o o o o • • • • o-o- o o o o o o o o o o o o • • o 1 o o • • • • • A o o o o • • • o 1 1 o 1 o 1 ? o A • • • • • o o o o o o o o I o 1 o 1 ? o A • • • • • A o o o o o o o o A • • • • • • A o o o o o o 1 o 1 o 1 o A • • • • • • • A o o 0 o o 1 o 1 ? o A • • • • • • • • A A o o o 1 o 1 o A • • • • • • • • • • A A A 1 • I 1 o- o--o--o--0--o- -o - O - -0--o - 0-•0-0-o- o-•o- 1 -o ( I I I ( T i l ( i n ( TT I ( T i l ( M l M I 0 1 C I I 0 3 I I T 03 (T T 0 ] COT I 1 C O M ] tO I I ] ( O i l ] II o n rT o n [ToT ] l I OT] F i g u r e 9.5 S l i p mode d i s t r i b u t i o n i n the ( 0 1 0 ) p l a n e f o r the 45° cone a n g l e c r y s t a l c o r r e s p o n d i n g t o the MRSS d i s t r i b u t i o n shown i n F i g u r e 9 . 2 ( c ) . 135 Midway between the c e n t r e and t h e edge o f the w a f e r t h e r e i s a r i n g w i t h minimum s t r e s s e s w h i c h e x h i b i t s c i r c u l a r symmetry. At t h e edge o f the w a f e r the s t r e s s e s i n c r e a s e t o a maximum v a l u e . The s t r e s s a t t h e edge i s f i v e t i m e s l a r g e r t h a n th e s t r e s s at t h e c e n t r e . T h i s complex p a t t e r n i s c o n s i s t e n t w i t h what i s o b s e r v e d i n F i g u r e 9.6(a) a t 7.5 mm from t h e i n t e r f a c e . The r i n g a t t h e c e n t e r w i t h h i g h e r s t r e s s e s i s due t o t h e i r r e g u l a r c o n t o u r s i n t h e upper h a l f of t h e c r y s t a l , at A i n F i g u r e 9 . 6 ( a ) . F o r h i g h e r cone a n g l e s the s t r e s s d i s t r i b u t i o n i s s i m p l e r . The c r y s t a l s w i t h 30° and 4 5 ° cones shown i n F i g u r e 9.7(b) and ( c ) , e x h i b i t a l m o s t c i r c u l a r symmetry w i t h low s t r e s s e s i n t h e c e n t r a l a r e a w h i c h i n c l u d e s h a l f t h e t o t a l s u r f a c e a r e a . O u t s i d e t h i s r e g i o n the s t r e s s r a p i d l y i n c r e a s e s t o w a r d s the edge o f t h e w a f e r . F o r b o t h cone a n g l e s t h e s t r e s s e s at t h e edge a r e about f i v e t i m e s l a r g e r t han a t the c e n t r e . F o r l a r g e r cone a n g l e s a r e g i o n of h i g h e r s t r e s s e s s t a r t s t o b u i l d up a t the c e n t r a l r e g i o n , as shown i n F i g u r e 9.7(d) and ( e ) , p r o d u c i n g t h e c h a r a c t e r i s t i c W-shaped s t r e s s d i s t r i b u t i o n a c r o s s t h e d i a m e t e r . Fo r the l a r g e s t cone a n g l e (65°) t h e s t r e s s e s a t t h e c e n t r e a r e t h r e e t i m e s l a r g e r t h a n the m i n i m a l s t r e s s e s a t t h e mid r a d i u s p o s i t i o n . At t h e edge the s t r e s s e s a r e a bout e i g h t t i m e s l a r g e r t h a n th e minimum v a l u e s . I t i s shown l a t e r t h a t t h e q u a l i t a t i v e c h a r a c t e r i s t i c s o f symmetry p r e s e n t e d h e r e a r e g e n e r a l . T h i s i s b e c a u s e the symmetry p a t t e r n s o f the s t r e s s f i e l d s a r e i n d e p e n d e n t of t h e t h e r m a l g r a d i e n t s , p r o v i d i n g 136 t h e t e m p e r a t u r e p r o f i l e r e m a i n s l i n e a r and the g r o w t h c o n d i t i o n s ( r a d i u s , b o r o n o x i d e t h i c k n e s s , e t c . ) r e m a i n unchanged. The d e n s i t y and d i s t r i b u t i o n o f d i s l o c a t i o n s i n t h e c r y s t a l i s r e l a t e d to t h e l o c a l e x c e s s s t r e s s above the c r i t i c a l r e s o l v e d s h e a r s t r e s s (MRSS-CRSS). When the CRSS ( Y i e l d ) i s t a k e n as the CRSS, the r e s i d u a l s t r e s s depends on t h e cone a n g l e . F o r t h e s m a l l e s t cone a n g l e the MRSS i s a l w a y s l a r g e r t h a n the CRSS ( Y i e l d ) and the symmetry i s v e r y s i m i l a r t o t h e symmetry o f the MRSS as shown i n F i g u r e 9 . 6 ( a ) . F o r l a r g e r cone a n g l e s th e MRSS i n e x c e s s o f t h e CRSS ( Y i e l d ) i s g r e a t e r t h a n z e r o i n t h e c e n t r a l and p e r i p h e r a l r e g i o n s i n the w a f e r . T h i s i s shown i n F i g u r e 9.8 f o r t h e 6 5 ° cone a n g l e . S i m i l a r r e s u l t s a r e o b t a i n e d f o r t h e o t h e r cone a n g l e s . The s t r e s s d i s t r i b u t i o n s on c r o s s - s e c t i o n s 2.5 mm from the i n t e r f a c e a r e shown f o r cone a n g l e s 7.1° ( F i g u r e 9.9(a-b) and 4 5 ° ( F i g u r e 9 . 9 ( c ) ) . In F i g u r e 9.9(a) the MRSS d i s t r i b u t i o n e x h i b i t s a W shape a c r o s s th e d i a m e t e r and has e i g h t - f o l d symmetry i n t h e c e n t r e . The MRSS-CRSS ( Y i e l d ) ( F i g u r e 9 . 9 ( b ) ) i n t h e c e n t r a l r e g i o n i s below z e r o so t h a t no d i s l o c a t i o n s or symmetry p a t t e r n s w i l l be e v i d e n t . As the cone a n g l e i n c r e a s e s t h e e i g h t - f o l d symmetry i n the c e n t r e i s r e p l a c e d by f o u r - f o l d symmetry as shown i n F i g u r e 9.9(c) f o r a 45° cone a n g l e . In t h i s c a s e t h e d i s l o c a t i o n symmetry w i l l be e v i d e n t . I t can a l s o be o b s e r v e d i n t h i s f i g u r e t h a t t h e c e n t r e has s t r e s s l e v e l s t h r e e t i m e s l a r g e r t h a n the edge. The s t r e s s d i s t r i b u t i o n a l o n g th e r a d i u s depends on t h e d i r e c t i o n . In the [110] d i r e c t i o n s t h e s t r e s s s t a r t s t o 137 i g u r e 9.6 C o n t o u r s i n MPa of the MRSS i n e x c e s s o f (a) CRSS ( Y i e l d ) ; (b) CRSS (MB) and ( c ) CRSS (MBTe). In most of the c r y s t a l the MRSS i s l a r g e r t h a n the c r i t i c a l v a l u e s . The bump shaped c o n t o u r s (A) g i v e complex s t r e s s d i s t r i b u t i o n s i n p e r p e n d i c u l a r c r o s s - s e c t i o n s . 138 a 139 140 C 141 d g u r e 9.7 MRSS c o n t o u r s (MPa) i n c r o s s - s e c t i o n s p e r p e n d i c u l a r t o t h e c r y s t a l a x i s f o r f i v e cone a n g l e s . (a) 7 . 1 ° , (b) 30°, ( c ) 4 5 ° , (d) 54. 7 ° , (e) 65 . S e c t i o n (a-d) a r e 7.5 mm from t h e i n t e r f a c e and s e c t i o n (e) i s 8.0 mm from the i n t e r f a c e . The h o r i z o n t a l d i r e c t i o n c o r r e s p o n d s t o t h e [100] d i r e c t i o n and the v e r t i c a l d i r e c t i o n c o r r e s p o n d s t o t h e [010] d i r e c t i o n . 142 F i g u r e 9.8 C o n t o u r s of t h e MRSS-CRSS ( Y i e l d ) (MPa) f o r the 65° cone a n g l e c r y s t a l . MRSS c o n t o u r s a r e shown i n F i g u r e 9 . 7 ( e ) . At h i g h cone a n g l e s MRSS l e v e l s a r e l a r g e r t h a n CRSS at the c e n t r e and o u t s i d e p a r t o f the w a f e r . a 144 145 F i g u r e 9.9 S t r e s s c o n t o u r s (MPa) i n c r o s s - s e c t i o n 2.5 mm from t h e i n t e r f a c e . (a) and (b) i n a 7.1 cone a n g l e c r y s t a l , ( c ) i n a 45 cone a n g l e c r y s t a l , (a) MRSS c o n t o u r s , (b) and ( c ) MRSS-CRSS ( Y i e l d ) . F o r the 7.1° cone a n g l e t h e r e i s e i g h t - f o l d symmetry a t the c e n t r e i n (a) which i s not seen i n (b) b e c a u s e s t r e s s l e v e l s a r e l e s s t h a n CRSS. F o r the 45° cone a n g l e t h e r e i s f o u r - f o l d symmetry. 146 d r o p c l o s e r to the c e n t e r t h a n i n any o t h e r d i r e c t i o n ; i n a d d i t i o n t h e s t r e s s r e m a i n s low even c l o s e t o t h e edge. As a r e s u l t l a r g e r a r e a s o f low s t r e s s e s a r e p r e s e n t i n the <110> d i r e c t i o n s . S i m i l a r c o n c l u s i o n s were o b t a i n e d w i t h the h i g h e r cone a n g l e s . In a l l the c a s e s c o n s i d e r e d t h e r e i s one common c h a r a c t e r i s t i c p r e s e n t . At the edge o f t h e w a f e r e i g h t - f o l d symmetry i s always p r e s e n t w i t h minima i n t h e <100> and <110> d i r e c t i o n s . T h i s i s more a p p a r e n t i n F i g u r e 9 . 7 ( d ) , where the maxima a r e s y m m e t r i c a l l y d i s t r i b u t e d . The e i g h t - f o l d symmetry a t th e edge i s a s s o c i a t e d w i t h an e i g h t - f o l d d i s t r i b u t i o n of the s l i p mode. T h i s i s shown i n F i g u r e 9.10. I t i s a l s o o b s e r v e d t h a t t h e l o c a l minima a t t h e edge i n the <100> and <110> d i r e c t i o n s a r e a s s o c i a t e d w i t h a change i n t h e s l i p mode. 9.1.2 E f f e c t of T h e r m a l C o n d i t i o n s In o r d e r to a s s e s s t h e e f f e c t of the t h e r m a l c o n d i t i o n s on the s t r e s s d i s t r i b u t i o n d u r i n g growth, s t r e s s f i e l d s a r e d e t e r m i n e d i n c r y s t a l s h a v i n g a 4 5 ° cone a n g l e f o r d i f f e r e n t t h e r m a l g r a d i e n t s i n the b o r o n o x i d e and a r g o n g as, as l i s t e d i n T a b l e 9.2 . The c a l c u l a t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e g r o w i n g c r y s - t a l f o r a range of t h e r m a l g r a d i e n t s i s shown i n F i g u r e 9.11(a- d ) . The i s o t h e r m s a r e o b s e r v e d t o be v e r y f l a t f o r low t h e r m a l g r a d i e n t s w i t h a s m a l l c u r v a t u r e a t t h e s h o u l d e r of the c r y s t a l . 147 o ( 1 " " I ) [ 1 1 0 1 0 ( 1 1 i ) L O T 1 ] • ( T " " 1 ) 10 I I D ft ( I 1 ) L 1 OT 3 • ( T i n [ T O T ] * ( i T I ) 11 o f ] B ( I 7 I ) C O I I 3 e ( 7 I I ) [ OT I 3 C 0 I 0 3 • ( I I I ) ( I I I ) ( T i l ) ( T T I ) [ 1 0 13 [ T I 03 [ M O D t I T 03 ^ f t * ft ft A * * A . * ft /° ° * * ft ^ ° ° ° • ft A A ft s a o a a a a D f t A B # # /  D ° D O D A ( S • • • • • O f t ft ft ft ft ft o o a A • • • • • • • f t fta 99 ft ft ft ft o o o s ft • • • • • • • 4 * 8 9 9 9 9 o o o o o • ft A ft 9 ft O O o • • ft ft ft ft • • • ft ft a a \ a a a a a^ a a a a a a ° o • • " - \ # Q 9 a a B B a a a ^ ft • • a a a a a a a 0 0 • • • a a a a a a a - ^ • • • • • • • f t f t B 9 9 9 9 0 0 0 e « d O A f t » B e \ 9 0 0 0 9 0 0 f t f t < * B 999Z99**9 9 \ a 0 0 0 0 0 * * * * 9 9 e 0 0 0 O 9 o o o a a a a a 9 f t • • • • • O O O O O O 9 9 9 [ | 0 0 3 0 B O O O 0 0 0 9 0 > • 0 0 9 9 9 9 9 9 B • 9 9 9 9 9 9 9 • » » 9 0 9 9 9^ • • 9 9 9 9 9 ^ * • 9 9 9 J # • • 9 • 0 . V a • 9 • • ft • • • • a • a ft A ft • • • • ft ft 9 • • • ft ft • A ft • 0 ft • ft A • • 0 9 0 9 9 9 0 # 9 9 9 9 9 9 ft ft ft ft ft m * ft ft ft ft 9 ft ft ft 9 9 ft ft ft 9 9 ft ft ft 9 9 ft ft ft 9 9 * • 0 F i g u r e 9.10 S l i p mode d i s t r i b u t i o n i n a (001) p l a n e c o r r e s p o n d i n g t o t h e MRSS d i s t r i b u t i o n shown i n F i g u r e 9 . 7 ( c ) f o r the 45 cone a n g l e c r y s t a l a t 7.5 mm from the i n t e r f a c e . The e i g h t - f o l d d i s t r i b u t i o n o f t h e mode at t h e edge o f t h e s e c t i o n i s a s s o c i a t e d w i t h t h e e i g h t - f o l d symmetry of the MRSS. 148 a b 149 F i g u r e 9.11 T e m p e r a t u r e d i s t r i b u t i o n as a f u n c t i o n of e x t e r n a l t e m p e r a t u r e g r a d i e n t f o r a 4 5 ° cone a n g l e c r y s t a l . T h e r m a l g r a d i e n t s i n the b o r o n o x i d e a r e : (a) 50°C/cm, (b) 100°C/cm, ( c ) 200°C/cm and (d) 400°C/cm. T e m p e r a t u r e s a r e g i v e n i n 10 °C. F o r low g r a d i e n t s i s o t h e r m s a r e n e a r l y f l a t and s l i g h t l y convex. F o r l a r g e r g r a d i e n t s i s o t h e r m s a r e c u r v e d and c o n c a v e . 150 T a b l e 9.2 E f f e c t of Ambient T e m p e r a t u r e on A x i a l T e m p e r a t u r e G r a d i e n t and S t r e s s T e m p e r a t u r e G r a d i e n t s , °C/cm B o r o n o x i d e | A r g o n | A x i a l | AMRSS, MPa 50 25 25 0.59 100 50 48 2.55 200 100 84 5.12 400 100 98 11.91 F o r a t e m p e r a t u r e g r a d i e n t of 400°C/cm ( F i g u r e 9 . 1 1 ( d ) ) , the i s o t h e r m s a r e c l e a r l y c u r v e d w i t h the c u r v a t u r e i n c r e a s i n g near the s h o u l d e r . The f i g u r e a l s o shows t h a t t h e a x i a l t e m p e r a t u r e g r a d i e n t i s h i g h e r t h a n the r a d i a l g r a d i e n t . The a x i a l g r a d i e n t s i n t h e c r y s t a l f o r t h e d i f f e r e n t imposed g r a d i e n t s a r e l i s t e d i n T a b l e 9.2. I t can be s e e n t h a t t h e s e g r a d i e n t s a r e not l i n e a r l y r e l a t e d . In t h e same t a b l e t h e l a r g e s t MRSS (AMRSS), w h i c h o c c u r s below the s h o u l d e r , i s a l s o l i s t e d . The r e l a t i o n between the a x i a l g r a d i e n t and t h e AMRSS i s not l i n e a r . The s t r e s s d i s t r i b u t i o n i n the c r y s t a l g r o w i n g under a t h e r m a l g r a d i e n t of 50°C/cm i s shown i n F i g u r e 9.12(a) and ( b ) , f o r a 200°C/cm g r a d i e n t i n F i g u r e 9.13(a) and (b) and f o r a 400°C/cm g r a d i e n t i n F i g u r e 9.14(a) and ( b ) . P a r t (a) g i v e s t h e MRSS, p a r t (b) the MRSS-CRSS ( Y i e l d ) . At 50°C/cm ( F i g u r e 9 . 1 2 ( a ) ) the MRSS has a d i f f e r e n t s t r e s s d i s t r i b u t i o n t h a n a t h i g h e r g r a d i e n t s s p e c i f i c a l l y the h i g h s t r e s s below t h e s h o u l d e r 151 F i g u r e 9.12 S t r e s s c o n t o u r s (MPa) i n a C 0 1 0 ^ p l a n e f o r a 45° cone a n g l e c r y s t a l grown w i t h a 50 C/cm g r a d i e n t i n t h e bo r o n o x i d e , (a) MRSS c o n t o u r s does not show t h e bump shaped s t r e s s d i s t r i b u t i o n below t h e s h o u l d e r i n ( a ) . The MRSS-CRSS ( Y i e l d ) i s p o s i t i v e o n l y i n a few r e g i o n s i n ( b ) . 152 F i g u r e 9.13 (a) MRSS c o n t o u r s and (b) MRSS-CRSS ( Y i e l d ) c o n t o u r s f o r a 200 C/cm g r a d i e n t i n t h e b o r o n o x i d e . U n i t s a r e i n MPa. MRSS s t r e s s a r e l a r g e below t h e s h o u l d e r i n ( a ) . The MRSS-CRSS i s g r e a t e r t h a n z e r o i n most of the c r y s t a l i n ( b ) . 153 Q b 154 O-O -OO-O -O 6 o ft ft ft 6 9 o ft ft ft cb I I I o o ft ft a 6 6o ft ft D 6 6 O D O o o IV o o o o o o o o o 6 o o 6 o o 6 o o o o o o o o V o o o o o o o o o\>s o o o o o o os o o o o o o o o \ \ o o o o o o o o o o o o o o o o o o o o o o o o o ^ 6 o o O O O O O O O O O O D O O O O O O f t f t f t f t f t f t f t f t D O O O O ft ft ft ft ft A ft ft A ft ftND O O f t f t f t f t f t f t f t O O O O O O D ^ J \ 6 o o o o ft ft ft ft ft ft ft ft ft ft ft • o o • • • ft ft ft ft o o o o o • • 6 6 o • • • • ft ft o o o o o o o • o 6 o o • • • • ft ft o o o o o o o <!> 6 o • • • • • ft ft o o o o o o o o ? ° • • • • • • ft ft o o o o o o 6 o o • • • • • • • • ft ft o o o o ft o o 1 o • • • • • • D • Q ft ft ft O D O - f t . -ft-- f t - .ft-- f t - -ft. -ft--ft-• ft--ft. - o - - o - - o - - o - -o-6 F i g u r e 9.14 (a) MRSS c o n t o u r s . (b) MRSS-CRSS ( Y i e l d ) c o n t o u r s . ( c ) S l i p mode d i s t r i b u t i o n . G r a d i e n t i n t h e b o r o n o x i d e i s 400 C/cm. In (a) c o n t o u r s a r e s i m i l a r t o t h o s e o b t a i n e d f o r a 200°C/cm g r a d i e n t . S t r e s s l e v e l s have d o u b l e d . In (b) o n l y a s m a l l a r e a i n the see d d e v e l o p e d s t r e s s e s l e s s t h a n t h e CRSS. In ( c ) the mode d i s t r i b u t i o n i s s i m i l a r to t h a t shown i n F i g u r e 9.5 f o r a 100°c/cm g r a d i e n t . 155 d i s a p p e a r s . At the h i g h e r g r a d i e n t s , F i g u r e s 9.13(a) and 9.14(a) t h e MRSS c o n f i g u r a t i o n i s s i m i l a r . The MRSS l e v e l i n c r e a s e s as the t h e r m a l g r a d i e n t i n c r e a s e s . T h i s i s shown i n p a r t (b) of F i g u r e s 9.12 to 9.14. F o r a g r a d i e n t 50°c/cm a l m o s t t h e whole c r y s t a l s e c t i o n has MRSS-CRSS ( Y i e l d ) l e v e l s below z e r o . The s l i p mode wh i c h i s o p e r a t i v e at 400°C/cm i s t h e same to the mode which o p e r a t e s a t 100°C/cm as seen by c o m p a r i n g F i g u r e s 9 .14(c) and 9.5. From t h i s i t can be c o n c l u d e d t h a t s i m i l a r s l i p modes come from s i m i l a r s t r e s s c o n f i g u r a t i o n s and t h i s i s i n d e p e n d e n t of the v a l u e of t h e c o n s t a n t t e m p e r a t u r e g r a d i e n t . S i m i l a r s l i p modes a r e e x p e c t e d f o r s t r e s s d i s t r i b u t i o n s w i t h s i m i l a r r e l a t i o n among s t r e s s components. In a d d i t i o n t h e symmetry of the MRSS does not change when the t h e r m a l g r a d i e n t i s i n c r e a s e d from 100°C/cm t o 400°C/cm as shown by F i g u r e s 9 . 7 ( c ) and 9.15. 9.1.3 E f f e c t of the Heat T r a n s f e r C o e f f i c i e n t C o n d i t i o n s : BG, 100°C/cm ; AG, 50°C/cm In o r d e r to d e t e r m i n e how s e n s i t i v e t h e c a l c u l a t i o n s a r e t o the m a g n i t u d e of the h e a t t r a n s f e r c o e f f i c i e n t s , t h e c a l c u l a t i o n s were r e p e a t e d w i t h v a l u e s of the he a t t r a n s f e r c o e f f i c i e n t i n c r e a s e d and d e c r e a s e d by a f a c t o r of 1.3 r e s p e c t i v e l y . The r e s u l t s a r e shown i n F i g u r e 9.16 f o r the t e m p e r a t u r e p r o f i l e s a t the s u r f a c e of the c r y s t a l . F o u r p r o f i l e s a r e p r e s e n t e d ; t h r e e c o r r e s p o n d to the t h r e e d i f f e r e n t heat t r a n s f e r c o e f f i c i e n t s and the f o u r t h c o r r e s p o n d s t o t h e ambient t e m p e r a t u r e . The 156 CO 1 0 ] F i g u r e 9.15 MRSS c o n t o u r s i n MPa i n a c r o s s - s e c t i o n a t 7.5 mm from t h e i n t e r f a c e f o r t h e c r y s t a l shown i n F i g u r e 9 . 1 4 ( a ) . The symmetry i s s i m i l a r to t h a t shown i n F i g u r e 9.7(c) f o r a 100°C/cm g r a d i e n t . 157 t e m p e r a t u r e p r o f i l e s a t t h e s u r f a c e and c e n t r e l i n e , w h i c h a r e not shown, a r e v e r y s i m i l a r t o t h o s e i n F i g u r e 9.16. The l a r g e s t e f f e c t o f the change i n h e a t t r a n s f e r c o e f f i c i e n t i s o b s e r v e d at t h e se e d end. The change i n t e m p e r a t u r e f o l l o w s the change i n the hea t t r a n s f e r c o e f f i c i e n t . At t h e c r y s t a l / c o n e j u n c t i o n t h e t e m p e r a t u r e change i s s m a l l , a p p r o x i m a t e l y 5°C. At t h e se e d t h e e f f e c t i s more p r o n o u n c e d b e c a u s e the s e e d r a d i u s i s s m a l l and h e a t t r a n s f e r i s c o n t r o l l e d a t t h e s e e d s u r f a c e . In t h e b u l k c r y s t a l t h e r e i s mixed c o n t r o l at t he s u r f a c e and i n the b u l k m a t e r i a l . As a r e s u l t , a change i n he a t t r a n s f e r c o e f f i c i e n t does not s u b s t a n t i a l l y m o d i f y t h e t e m p e r a t u r e f i e l d . The s t r e s s f i e l d s a r e shown i n F i g u r e 9.17 and 9.18 f o r the two h e a t t r a n s f e r c o e f f i c i e n t s h / 1.3 and h X 1.3 r e s p e c t i v e l y , (a) c o r r e s p o n d s t o the MRSS d i s t r i b u t i o n and (b) c o r r e s p o n d s t o the MRSS-CRSS ( Y i e l d ) . C o m p aring t h e maximum v a l u e s of MRSS o c c u r r i n g below t h e s h o u l d e r , i t can be seen t h a t the s t r e s s v a r i e s o n l y a b o u t 20 * i n the whole span of h e a t t r a n s f e r c o e f f i c i e n t v a l u e s . The span r e p r e s e n t s 53 % o f t h e o r i g i n a l v a l u e s a t t h e c o r r e s p o n d i n g t e m p e r a t u r e . I n s i d e t h e c r y s t a l , c h a n g i n g the heat t r a n s f e r c o e f f i c i e n t has a s t r o n g e f f e c t on the s t r e s s f i e l d . Note t h a t the minimum v a l u e of MRSS i n c r e a s e s from 0.05 MPa t o 0.16 MPa. I n c r e a s i n g t h e h e a t t r a n s f e r c o e f f i c i e n t does not change t h e d i s l o c a t i o n d i s t r i b u t i o n ( F i g u r e 9.17(b) and F i g u r e 9 . 1 8 ( b ) ) . T h i s can be 158 0 I 2 3 Position Relative To Interface ( c m ) F i g u r e 9.16 T e m p e r a t u r e p r o f i l e s a t the s u r f a c e o f 45 cone a n g l e c r y s t a l s f o r t h r e e ( l i f e r e n t c o n d i t i o n s . Curve H c o r r e s p o n d s t o t h e t e m p e r a t u r e s c a l c u l a t e d u s i n g the o r i g i n a l h e a t t r a n s f e r c o e f f i c i e n t v a l u e s . In c u r v e s H / 1.3 and H x 1.3 t h e o r i g i n a l h e a t t r a n s f e r c o e f f i c i e n t v a l u e s were d i v i d e d and m u l t i p l i e d by 1.3 r e s p e c t i v e l y . 159 F i g u r e 9.17 S t r e s s c o n t o u r s i n MPa d e r i v e d from the t e m p e r a t u r e f i e l d o b t a i n e d u s i n g h e a t t r a n s f e r c o e f f i c i e n t v a l u e s 1.3 l a r g e r t han o r i g i n a l v a l u e s . (a) MRSS c o n t o u r s , (b) MRSS-CRSS ( Y i e l d ) . 160 F i g u r e 9.18 S t r e s s c o n t o u r s i n MPa f i e l d o b t a i n e d u s i n g h e a t 1.3 t i m e s s m a l l e r t h a n c o n t o u r s . (b) MRSS-CRSS 20 % a r e o b s e r v e d i n t h e (a) and F i g u r e 9 . 1 7 ( a ) . d e r i v e d from t h e t e m p e r a t u r e t r a n s f e r c o e f f i c i e n t v a l u e s o r i g i n a l v a l u e s . (a) MRSS ( Y i e l d ) . V a r i a t i o n s o f o n l y maximum MRSS v a l u e s between 161 a c c o u n t e d f o r by t h e i n c r e a s e i n the t h e r m a l g r a d i e n t s i n the c r y s t a l r e s u l t i n g from t h e i n c r e a s e i n h e a t t r a n s f e r c o e f f i c i e n t , w h i c h l e a d s to two o p p o s i t e e f f e c t s . These a r e t h e h i g h e r t h e r m a l s t r e s s e s and t h e h i g h e r CRSS i n the c r y s t a l r e s u l t i n g from the l o w e r t e m p e r a t u r e s . When b o t h e f f e c t s a r e c o n s i d e r e d t h e net r e s u l t of t h e i n c r e a s e i n h e a t t r a n s f e r c o e f f i c i e n t on the d i s l o c a t i o n d i s t r i b u t i o n i s n e g l i g i b l e 9.1.4 E f f e c t o f N o n - l i n e a r i t y i n t h e T e m p e r a t u r e P r o f i l e The t e m p e r a t u r e f i e l d i n t h e c r y s t a l depends a l m o s t e n t i r e l y on two p a r a m e t e r s , t h e h e a t t r a n s f e r c o e f f i c i e n t a t t h e c r y s t a l s u r f a c e and t h e t e m p e r a t u r e p r o f i l e i n the media s u r r o u n d i n g the c r y s t a l . These two v a r i a b l e s a r e i n t e r r e l a t e d and may depend on many f a c t o r s l i k e h e a t i n p u t i n t o the mold, c r y s t a l d i a m e t e r and gas p r e s s u r e . However when the t e m p e r a t u r e p r o f i l e , t h e h e i g h t of the b o r o n o x i d e l a y e r , t h e gas c o m p o s i t i o n and t h e p r e s s u r e a r e known, the h e a t t r a n s f e r i s d e t e r m i n e d at each p o i n t on the c r y s t a l s u r f a c e . T h i s h e a t t r a n s f e r c o e f f i c i e n t , t o g e t h e r w i t h the t e m p e r a t u r e p r o f i l e , c r y s t a l s d i m e n s i o n s , and growth v e l o c i t y , d e t e r m i n e the t e m p e r a t u r e f i e l d i n s i d e t h e c r y s t a l . In th e p r e v i o u s s e c t i o n i t was shown t h a t a 30 * v a r i a t i o n i n the h e a t t r a n s f e r c o e f f i c i e n t does not s u b s t a n t i a l l y a f f e c t the s t r e s s f i e l d . However i t was shown t h a t c h a n g i n g t h e t e m p e r a t u r e p r o f i l e i n t h e s u r r o u n d i n g media has an i m p o r t a n t e f f e c t . I n c r e a s i n g or d e c r e a s i n g t h e t h e r m a l g r a d i e n t m a r k e d l y changes the t h e r m a l s t r e s s e s . In t h o s e c a l c u l a t i o n s t h e t e m p e r a t u r e p r o f i l e i n b o t h t h e b o r o n o x i d e and the a r g o n gas was assumed to 162 v a r y l i n e a r l y w i t h d i s t a n c e from t h e s o l i d / l i q u i d i n t e r f a c e . T h i s may not be the c a s e , as shown by t h e t e m p e r a t u r e p r o f i l e s m easured a l o n g the c r y s t a l s u r f a c e . In o r d e r t o d e t e r m i n e t h e i m p o r t a n c e of n o n - l i n e a r i t y o f t h e imposed t e m p e r a t u r e p r o f i l e , t e m p e r a t u r e s and s t r e s s e s a r e c a l c u l a t e d f o r t h e f o u r growth c o n d i t i o n s l i s t e d i n T a b l e 9.3 T a b l e 9.3 E f f e c t of N o n - l i n e a r i t y o f the Ambient T e m p e r a t u r e P r o f i l e on S t r e s s T e m p e r a t u r e G r a d i e n t s , °C/cm 200, z < 1.0 100, z > 1.0 AT i n B O . Run 1 1 2 3 I B 2 0 3 | Ar g o n | °C 140 100 350 100 350 100 , z < 1.0 50, z > 1.0 50 200 45 , z < 1.0 90 , z > 1.0 180 Note Boron o x i d e t h i c k n e s s = 25 mm ; a r g o n p r e s s u r e = 30 atm. ; r a d i u s = 20 mm ; l e n g t h = 10 mm ; cone a n g l e = 30° . z i s the d i s t a n c e from the s o l i d / l i q u i d i n t e r f a c e . 163 The c a l c u l a t i o n s a r e d i v i d e d i n two s e t s 1, 2 and 3, 4 t a k i n g i n t o a c c o u n t the d i f f e r e n c e i n t e m p e r a t u r e a c r o s s t h e b o r o n o x i d e l a y e r . Each p a i r of c o n d i t i o n s g i v e s s i m i l a r a v e r a g e t e m p e r a t u r e g r a d i e n t s a c r o s s t h e e n c a p s u l a n t . The d i f f e r e n c e i s i n t h e t e m p e r a t u r e p r o f i l e . In 1 t h e g r a d i e n t i s c o n s t a n t , i n 2 and 3 t h e g r a d i e n t i s l a r g e r c l o s e t o t h e m e l t , and i n 4 t h e g r a d i e n t i s l a r g e r c l o s e to the e n c a p s u l a n t / g a s i n t e r f a c e . The MRSS f o r t h e f o u r c a s e s a r e g i v e n i n F i g u r e 9 . 1 9 ( a - d ) . C o m p a r i n g (a) and (b) i t i s seen t h a t t h e r e l a t i v e s t r e s s d i s t r i b u t i o n i n b o t h c a s e s a r e s i m i l a r . The AMRSS below t h e s h o u l d e r f o r ca s e 2 i n (b) i s t w i c e as l a r g e as t h e e q u i v a l e n t s t r e s s f o r 1 i n ( a ) . The minimum MRSS i n s i d e t h e c r y s t a l i n (b) i s a b o u t 20 t i m e s l a r g e r t h a n i n ( a ) . C o m p a r i n g 3 and 4 i n F i g u r e 9.19 t h e MRSS d i s t r i b u t i o n s d i f f e r . In (d) t h e s t r e s s c o n c e n t r a t i o n o c c u r s a l o n g the cone s u r f a c e r a t h e r t h a n below t h e s h o u l d e r as i n c a s e s 1 t o 3. The AMRSS on t h e cone s u r f a c e i n (d) i s a bout 1.6 t i m e s l a r g e r t h a n i n ( c ) f o r 3. The l o w e s t s t r e s s i n (c ) o c c u r s a t two p o s i t i o n s , i n t h e cone and below t h e cone. In F i g u r e 9.19(d) the l o w e s t s t r e s s o c c u r s o n l y below the cone. The v a l u e of t h e minimum MRSS i n ( c ) and (d) i s t h e same. The symmetry of the s t r e s s e s i n (001) p l a n e s p e r p e n d i c u l a r t o t h e growth d i r e c t i o n i s s i m i l a r f o r 1, 2 and 3 o f T a b l e 9.3 and s i m i l a r to the symmetry o b t a i n e d f o r c r y s t a l s grown w i t h a 1 cm deep e n c a p s u l a n t and a g r a d i e n t of 100°C/cm. These symmetry p a t t e r n s a r e shown i n F i g u r e 9 . 7 ( c ) and 9 . 9 ( c ) . The symmetry o f th e s t r e s s e s f o r 4, however, i s f o u r - f o l d a t s e c t i o n s 5.0 and 164 165 F i g u r e 9.19 MRSS (MPa) c o n t o u r s f o r a c r y s t a l g r o w i n g under f o u r d i f f e r e n t c o n d i t i o n s g i v e n i n T a b l e 9.3. (a) Run #1 ; (b) Run #2 ; (c) Run #3 ; (d) Run #4. From (a) to (b) the AMRSS d o u b l e s . In (d) t h e bump shape below t h e s h o u l d e r does not a p p e a r . 166 7.5 mm from the i n t e r f a c e . At 5.0 mm e v i d e n c e of t w o - f o l d symmetry i s o b s e r v e d as shown i n F i g u r e 9.20. The MRSS-CRSS ( Y i e l d ) f o r c o n d i t i o n s 1 t o 4 a r e shown i n F i g u r e 9 . 2 1 ( a - d ) . F o r 1 t h e s t r e s s o v e r l a r g e a r e a s of the c r y s t a l i s l e s s t h a n the CRSS, f o r 2 o n l y s m a l l a r e a s have s t r e s s e s below the CRSS. S i m i l a r r e s u l t s a r e o b t a i n e d f o r 3 and 4. In 4, however, the r e g i o n w i t h z e r o s t r e s s a t the cone has moved t o a p o s i t i o n below th e cone. When c o m p a r i n g the s t r e s s e s o b t a i n e d f o r a b o r o n o x i d e l a y e r 2.5 cm t h i c k ( T a b l e 9.3) w i t h s t r e s s e s g e n e r a t e d w i t h a 1 cm l a y e r , t h e MRSS-CRSS ( Y i e l d ) f o r 1 i s c o m p a r a b l e to the d i s t r i b u t i o n o b t a i n e d u s i n g a g r a d i e n t of 100°C/cm a c r o s s t h e e n c a p s u l a n t ( F i g u r e 9 . 4 ( a ) ) . I t i s n o t e d t h a t i n 1 l a r g e r g r a d i e n t s a r e p r e s e n t , 40 * l a r g e r i n t h e c r y s t a l and 300 * l a r g e r i n the cone when compared to the g r a d i e n t s i n a 1 cm t h i c k l a y e r . T h i s r e s u l t shows the i m p o r t a n c e o f s u p p r e s s i n g t h e t h e r m a l d i s c o n t i n u i t y o c c u r i n g w i t h the 1 cm t h i c k e n c a p s u l a n t . The s t r e s s d i s t r i b u t i o n of F i g u r e 9.21(b) f o r case 2 i s c o m p a r a b l e to the d i s t r i b u t i o n i n F i g u r e 9.13(b) f o r a 1 cm t h i c k l a y e r . Comparable d i s t r i b u t i o n s a r e a l s o o b t a i n e d i n F i g u r e 9.21(c) f o r c a s e 3 and F i g u r e 9.4(a) f o r a 1 cm l a y e r . T h i s shows t h a t f o r t h i s c r y s t a l l e n g t h , i n c r e a s i n g t h e d e p t h of the b o r o n o x i d e from 1 to 2.5 cm w i t h the same t e m p e r a t u r e p r o f i l e i n the ambient does not d e c r e a s e t h e d i s l o c a t i o n d e n s i t y . In a d d i t i o n i t can be o b s e r v e d t h a t t h e a r e a s w i t h s t r e s s e s l e s s t h an the CRSS ( Y i e l d ) i n F i g u r e 9.21(c) a r e s i m i l a r t o t h o s e i n F i g u r e 9 . 2 1 ( a ) . 167 CO 10 ] F i g u r e 9.20 MRSS (MPa) c o n t o u r s i n a (001) p l a n e a t a d i s t a n c e of 5.0 mm from the i n t e r f a c e i n t h e c r y s t a l shown i n F i g u r e 9.19(d) ( r u n #4). The f o u r - f o l d symmetry w i t h a s l i g h t t w o - f o l d symmetry i s o b s e r v e d . 168 169 c d 21 MRSS-CRSS ( Y i e l d ) c o n t o u r s (MPa) f o r the f o u r g r o w th c o n d i t i o n s shown i n T a b l e 9.3 and MRSS c o n t o u r s shown i n F i g u r e 9.19 (a) Run #1 ; (b) Run #2 ; (c) Run #3 ; (d) Run #4 . 170 In 1, however, the g r a d i e n t Is a l m o s t t w i c e as l a r g e as the a v e r a g e g r a d i e n t i n 3. T h i s shows the d e t r i m e n t a l e f f e c t on s t r e s s of g r a d i e n t s w h i c h a r e not c o n s t a n t . From the r e s u l t s p r e s e n t e d i n t h i s s e c t i o n i t has been shown t h a t the e f f e c t of t h e shape o f the imposed t e m p e r a t u r e p r o f i l e i s v e r y i m p o r t a n t and t h a t t h e a p p r o x i m a t i o n of a c o n s t a n t g r a d i e n t i n the t e m p e r a t u r e p r o f i l e i n the b o r o n o x i d e , as done 19 5 by Dusseaux , g i v e s s t r e s s e s w h i c h a r e d i f f e r e n t t h a n the s t r e s s e s o b t a i n e d u s i n g v a r i a b l e g r a d i e n t s . In a d d i t i o n i t has been shown t h a t i n c r e a s i n g t h e b o r o n o x i d e t h i c k n e s s a l o n e , w i t h o u t c o n s i d e r i n g t h e changes i n the t h e r m a l p r o f i l e s i n t h e s u r r o u n d i n g media, does not c o n t r i b u t e t o a s u b s t a n t i a l d e c r e a s e i n t h e s t r e s s l e v e l s . 9 . 2 C r y s t a l L e n g t h C o n d i t i o n s R, 27.5 mm ; B, 21.0 mm CL : a) 13.75 mm ; b) 27.5 mm ; c) 55.0 mm ; d) 82.5 mm ; e) 110 mm . In the p r e v i o u s s e c t i o n i t was shown t h a t the a m bient t e m p e r a t u r e p r o f i l e i s a v e r y i m p o r t a n t v a r i a b l e i n d e t e r m i n i n g the s t r e s s l e v e l i n a g r o w i n g c r y s t a l . T h i s i s f o r b o t h t h e a x i a l t e m p e r a t u r e g r a d i e n t and t h e c u r v a t u r e of t h e t e m p e r a t u r e p r o f i l e . As a r e s u l t i t i s v e r y i m p o r t a n t to use, as i n p u t i n the model, t e m p e r a t u r e p r o f i l e s w h i c h a r e as c l o s e as p o s s i b l e t o r e a l c o n d i t i o n s i n s i d e t h e c r y s t a l growth chamber. Measurements 171 o f t e m p e r a t u r e p r o f i l e s d u r i n g g r o w t h have been r e p o r t e d u s i n g t h e t h e r m o c o u p l e a r r a n g e m e n t shown i n F i g u r e 8.18. The r e s u l t s a r e shown i n F i g u r e 8.19 i n wh i c h c u r v e 4 g i v e s t h e ambient t e m p e r a t u r e a d j a c e n t t o t h e c r y s t a l s u r f a c e . The MRSS d i s t r i b u t i o n f o r t h e f i v e c r y s t a l l e n g t h s c o n s i d e r e d a r e shown i n F i g u r e 9 . 2 2 ( a - e ) . Comparing d i s t r i b u t i o n s shows the f o l l o w i n g . 1) In a l l c a s e s h i g h s t r e s s l e v e l s o c c u r a t the c e n t r e and the o u t s i d e p a r t o f t h e c r y s t a l . F o r t h e s h o r t e s t c r y s t a l , (a) h i g h s t r e s s e s a r e a l s o p r e s e n t c l o s e t o the s o l i d - l i q u i d i n t e r f a c e . 2) The l o c a t i o n of t h e maximum (AMRSS) i s alw a y s a t t h e c r y s t a l s u r f a c e and i t s p o s i t i o n i n t h e a x i a l d i r e c t i o n c hanges w i t h c r y s t a l l e n g t h . F o r c r y s t a l l e n g t h s l e s s t h a n t h e t h i c k n e s s o f t h e e n c a p s u l a n t the p o s i t i o n o f th e AMRSS i s below t h e s h o u l d e r . F o r l o n g e r c r y s t a l s the AMRSS i s above t h e e n c a p s u l a n t - g a s s u r f a c e . 3) The minimum s t r e s s e s o c c u r a t b o t h ends of the c r y s t a l and a l s o i n a narrow a x i a l band a l o n g t h e c r y s t a l , u s u a l l y h a l f way between t h e c e n t r e and o u t s i d e p a r t o f the c r y s t a l . The p o s i t i o n o f the minimum moves t o the c r y s t a l s u r f a c e a t b o t h ends. The s t r e s s d i s t r i b u t i o n i n t h e r a d i a l d i r e c t i o n a lways e x h i b i t s a W shape . C l o s e t o t h e ends t h e v a l l e y s i n the W a r e c l o s e r t o the o u t s i d e p a r t of t h e c r y s t a l .  173 4) The v a l u e of the AMRSS which i s r e p r e s e n t a t i v e of the s t r e s s l e v e l i n t h e whole c r y s t a l , i n c r e a s e s w i t h c r y s t a l l e n g t h , r e a c h i n g a maximum v a l u e f o r ( c ) , and s l i g h t l y d e c r e a s e s f o r l o n g e r c r y s t a l s . The maximum AMRSS f o r ( c ) , f o r a c r y s t a l l e n g t h of 55.0 mm ( a s p e c t r a t i o = d i a m e t e r / l e n g t h = 1) i s t h e r e s u l t of two main f a c t o r s . F i r s t , h i g h e r s t r e s s e s a r e e x p e c t e d above the e n c a p s u l a n t s u r f a c e b e c a u s e of the d i s c o n t i n u i t y i n t h e media s u r r o u n d i n g t h e c r y s t a l . T h i s p r o d u c e s a sudden change i n h e a t t r a n s f e r c o e f f i c i e n t and g i v e s a maximum c u r v a t u r e i n the t e m p e r a t u r e p r o f i l e i n the s u r r o u n d i n g media. In F i g u r e 8.19 c u r v e 4, t h i s i s o b s e r v e d a t 30 mm from the i n t e r f a c e . As a r e s u l t of t h e s e changes i n the t e m p e r a t u r e f i e l d t h e s t r e s s e s a r e h i g h . The s e c o n d f a c t o r c o n t r i b u t i n g to the l a r g e s t AMRSS f o r ( c ) i s r e l a t e d t o t h e geometry o f t h e d e f o r m a t i o n . F o r t h e 55 mm c r y s t a l l e n g t h t h e e f f e c t of the t h e r m a l t r a n s i t i o n o c c u r s a t t h e m i d l e n g t h of the c r y s t a l . The m a t e r i a l i n t h i s r e g i o n i s more r e s t r a i n e d t o d e f o r m t h a n t h e r e g i o n s c l o s e t o t h e ends. T h i s q u a s i - p l a n e s t r a i n d e f o r m a t i o n g i v e s h i g h e r s t r e s s e s . The c o n t o u r s of t h e MRSS-CRSS ( Y i e l d ) f o r the f i v e c r y s t a l o f i n c r e a s i n g l e n g t h s a r e shown i n F i g u r e 9 . 2 3 ( a - e ) . The r e s u l t s a r e s i m i l a r t o t h e MRSS c o n s i d e r e d p r e v i o u s l y . In a d d i t i o n i t can be o b s e r v e d t h a t a t the s h o r t e s t l e n g t h f o r (a) t h e r e i s a s m a l l a r e a i n t h e cone as w e l l as below t h e cone where t h e MRSS i s l e s s t h a n the CRSS ( Y i e l d ) . F o r t h e two l a r g e s t c r y s t a l s ( ( d ) and ( e ) )  175 t h e top of the c r y s t a l does not d e v e l o p s t r e s s e s h i g h e r than the CRSS ( Y i e l d ) . When the CRSS (MB) s t r e s s e s f o r d i s l o c a t i o n g e n e r a t i o n i n undoped GaAs i s c o n s i d e r e d , t h e c o n t o u r s and s t r e s s v a l u e s a r e v e r y s i m i l a r t o t h o s e o b t a i n e d f o r t h e MRSS ( F i g u r e 9.22). The c o n t o u r s of the MRSS-CRSS (MBTe) i n T e - d o p e d m a t e r i a l a r e s i m i l a r t o t h o s e o b t a i n e d when the y i e l d s t r e s s i s c o n s i d e r e d and shown i n F i g u r e 9.23. The o p e r a t i v e s l i p mode of t h e MRSS as a f u n c t i o n of p o s i t i o n i n t h e c r y s t a l i s shown i n F i g u r e 9 . 2 4 ( a - e ) . The d i s t r i b u t i o n i n d i c a t e s t h a t t h e s l i p mode o p e r a t i n g may not be a s s o c i a t e d w i t h t h e s t r e s s l e v e l . The s l i p mode a l t e r n a t i v e l y may be r e l a t e d t o s p e c i f i c r e g i o n s i n t h e c r y s t a l . The most f r e q u e n t modes a r e I I I and IV. Mode I I I a p p e a r s i n t h e a r e a of maximum s t r e s s above t h e e n c a p s u l a n t s u r f a c e and at t h e c e n t r e of the c r y s t a l . I t a l s o a p p e a r s i n the cone and i n t h e c e n t r a l r e g i o n a t a d i s t a n c e from th e i n t e r f a c e g r e a t e r t h a n 27.5 mm. Mode IV p r e f e r e n t i a l l y a p p e a r s i n the f o l l o w i n g r e g i o n s (a) c l o s e to t h e i n t e r f a c e (b) above t h e r e g i o n of maximum s t r e s s e s and ( c ) below the s h o u l d e r . The o t h e r s l i p modes a r e l e s s f r e q u e n t . 9.2.1 E f f e c t o f c r y s t a l L e n g t h on S t r e s s Symmetry The e f f e c t of c r y s t a l l e n g t h on s t r e s s symmetry i s p r e s e n t e d i n two ways. F i r s t , d i f f e r e n t c r o s s - s e c t i o n s a r e f o l l o w e d d u r i n g g rowth and t h e symmetry changes a r e d e s c r i b e d . F o l l o w i n g t h i s the symmetry a t each c r y s t a l l e n g t h i s c o n s i d e r e d and t h e changes w i t h l e n g t h a r e r e p o r t e d . 176 •-D-O-D-O-O-B-B - B - B - B - B - B - f l - B - B - B - B -O-O-O-O-O-O-O-O-B-B-^-^-^-O-O / • • • • • • • • • • • • • • • • • • • • • • • • a a a o a o o o o o o o o ^ ^ o ^ i / . O D a Q O a a o a D i i i i i i n i a Q Q G Q a Q D Q D o o o o o o o o ^ a a ^ o 0 o 4 4 4 4 4 4 ^ D « « t * * « « * * * « « « o 4 4 4 4 4 < l * * « * o o o o o o o 4 _ /o 0 o o o 4 4 o o o o o o o o o o o o o o o o o o o o o B B B B B B a a a a o a a o « / ° ° 0 O O 0 O O O 0 0 0 0 O 0 O 0 O O 0 O 0 0 o 0 O 0 0 0 O B B B D D O D a D D D D 0 » o 0 ° 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o i D D D D o a o a a D o i O o 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o a r j a r j a a a o a a a « / o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o a a o D D D O O o o i • - • o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o i o-o-o-o-o-o-o-o-o-o-o-o-o-o-o- o - o - o - o - o - o - o - o - o - o - o - o - o - o - o - o - o - o - o - o o - o - o o - o - o - o - o - o- o - c - o - 6 D-Q-O-O-O / * • • • • • o o • • a a ' o _ o o o o o w O n O / O o o 0 0 0 0 0 p 0 g 0 ° ° o o o o S ° o o O o o O O / o o ° o ° ° 0 0 0 0 • " • 0 0 0 0 0 0 0 0 OO 0 - 0 - 0 0 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 B-B-B-B-B-B-B-B-B-O-O-O-O-O-O-O-O-O-O- • • • • • • • • • • a o a o o o o o o • • o o d d o o o a o o r j o o o o o o O O O O O O O O O O I I I I I I I Q Q O O O O O O O O O O I I I I D E I D D O O O O O O O O O O O I I O O D O D O O O O O O O O O O O O D D D O D A O O O O O O O O O O O O O Q D D Q O O O O O O O O O O O O O O O O O O O 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 0 - 0 - 0 - 0 - 0 - 0 - a -o-4-4 -a-o 0 0 o 4 4 o 4 O O O < J • • • o o ^ a a c ^ • a • a • a • o a • • a • f O O O O O O f • 0 - 0 - 0 - 0 - 0 - 0 - 0 a - a - a - D - o - o - o - o - D - o - o - o - o - o - o - o ^ -4 <a-o Q/D D Q G Q Q O Q D 0 0 0 0 0 0 0 0 0 4 0 4 / Q 4 • • • • • • • • o o o o o o o o o a a * p0m<4<4M44M •••••••• oaa • ^ " Q O O O O O I I I I I I I I Q D D D D D D I 0 O a 0 0 O 0 0 0 0 0 I I I I I Q D C i a c i c l D C 1 * / o Q ° 0 o o o o o o o o i i i a o a a a a Q a o i / O O O O O O O O O O O O O O I I I O D D O D Q Q Q I •-o o o o o o o o o o o o o o o o o o o m i o o o i 6 - 0 - 0 - 0 - 0 - 0 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - o - o - o - o - 0 - 0 - 0 - 0 - 0 - 0 •-a-0-0-0 -O-0-4-4 - a - o Ja • • D 0 0 0 0 0 0 4 o 0 o o o n i i i D a a Q « 0 ^ O o O O B B B B B Q D D n » / O o ° O O O O B B B B B B B D » • - • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6-000-0000-00-0-0-0-0-0-0-0-6 D-O-O^-O-O - - •— •— 1 — 1 — — •— o o o o 0 0 I — I — - I— I — I - I — - I — I- '• O O • B / • • D » • B • O o o « » B D / 0 Q Q « « B B / J O B O O B O B B . / J Q O B O O B O O B • • - • O o O B O O O O O O O O OOOO-OOO-O-O-O-O-O • > O o o o — - I— — — I— 0 0 — I - I — I - CO 1—4 X CO •*-> £ 4-> to E CO c >> cu i n u 1—4 0 c - CM CO • 4-> -̂̂  CO E >> •—• E Si 0 O cu O > --1 -1-4 i-i E E ^—- cu i n J= t> 4-> CO -- (1 1-1 O £ tM .—. E CO ' IO CO t o 00 as CO co s CM • . E cu - — E cu +-> u lO 3 to t> 0 E CM b E cu CO C O •1-1 O -11 • e CO c m 3 a 2 • r t 1—( O T 3 —H X CO 03 CO U CO C7> CU Si 3 to •—) 177 C o n s i d e r i n g the s t r e s s symmetry d u r i n g growth f o u r s e c t i o n s f i x e d i n t h e c r y s t a l a r e examined a t 8.25 mm, 49.5 mm, 77.0 mm and 104.5 mm from the cone. The symmetry and s t r e s s l e v e l s i n t h e s e c r o s s - s e c t i o n s a r e c o n s i d e r e d i n r e l a t i o n t o d i s l o c a t i o n f o r m a t i o n . In the model the d i s l o c a t i o n d e n s i t y formed a t any g i v e n t i m e and p o s i t i o n i s assumed t o be p r o p o r t i o n a l t o the MRSS-CRSS ( Y i e l d ) . However, t h e s t r e s s l e v e l changes d u r i n g g r o w t h . I t i s t h e r e f o r e i m p o r t a n t t o i n t r o d u c e two new c o n s i d e r a t i o n s . The f i r s t i s r e l a t e d to h a r d e n i n g . The e f f e c t of h a r d e n i n g i s to d e c r e a s e t h e e f f e c t i v e s t r e s s i n the c r y s t a l due to d i s l o c a t i o n i n t e r a c t i o n s . The d e c r e a s e i n s t r e s s i s p r o p o r t i o n a l t o t h e s q u a r e r o o t of t h e d i s l o c a t i o n d e n s i t y , i . e . A x /~Ii, where A i s the i n t e r a c t i o n c o n s t a n t and n i s the 19 2 d i s l o c a t i o n d e n s i t y . A c c o r d i n g t o Vakhrameev e t a l . A v a r i e s - 2 from 2 t o 4 x 10 MPa x m. The m a g n i t u d e of s t r e s s i n v o l v e d i n - 3 - 2 t h e i n t e r a c t i o n v a r i e s between 4 x 10 MPa and 4 x 10 MPa f o r 2 4 2 d i s l o c a t i o n d e n s i t i e s of 10 to 10 /cm These s t r e s s e s a r e n e g l i g i b l e w i t h r e s p e c t to the t h e r m o e l a s t i c s t r e s s e s d e v e l o p e d d u r i n g g r owth. T h e r e f o r e i t may be c o n c l u d e d t h a t the h a r d e n i n g e f f e c t i s m i n i m a l f o r normal d i s l o c a t i o n d e n s i t i e s . The s e c o n d c o n s i d e r a t i o n i s p a r t i a l l y a c o n s e q u e n c e of t h e f i r s t and i s r e l a t e d to the i n c r e a s e i n d i s l o c a t i o n s a f t e r e ach s t e p d u r i n g growth. Because t h e r e i s no h a r d e n i n g e f f e c t i t i s assumed t h a t the d i s l o c a t i o n d e n s i t y i n a g i v e n s e c t i o n i n c r e a s e s p r o p o r t i o n a l l y t o the d i f f e r e n c e i n MRSS-CRSS ( Y i e l d ) v a l u e s between two s t e p s . The f i n a l d e n s i t y i s g i v e n by the maximum s t r e s s r e a c h e d at each p o i n t . 178 The s p e c i f i c o b s e r v a t i o n s i n the f o u r s e c t i o n s f o l l o w . In a l l c a s e s the MRSS-CRSS ( Y i e l d ) i s c o n s i d e r e d . 1) S e c t i o n at 8.25 mm from the cone. CL : (a) 13.75 mm ; (b) 27.5 mm ; ( c ) 55.0 mm; (d) 82.5 ram. The s t r e s s d i s t i b u t i o n s a t the f o u r s t e p s a r e shown i n F i g u r e 9 . 2 5 ( a - d ) . The d i s t r i b u t i o n shown i n (a) c o r r e s p o n d s t o an e a r l y s t a g e of growth when the c r y s t a l i s c o n t a i n e d by the e n c a p s u l a n t . The s t r e s s l e v e l i n most of t h e c r y s t a l s e c t i o n i s l e s s t h a n "1 MPa. The symmetry changes d e p e n d i n g on t h e p o s i t i o n i n t h e s e c t i o n . C i r c u l a r symmetry i s o b t a i n e d i n t h e c e n t r e which changes t o f o u r - f o l d , e i g h t - f o l d , c i r c u l a r and f i n a l l y e i g h t - f o l d on moving from the c e n t r e to the edge of t h e c r y s t a l . In F i g u r e 9.25(a) the d i s l o c a t i o n d e n s i t y i s low and has t h e same symmetry. In (b) the s t r e s s l e v e l at t h e edge i s about 6 t i m e s h i g h e r than i n ( a ) . The changes a r e not as i m p o r t a n t at t h e c e n t r e where the s t r e s s has not q u i t e d o u b l e d . In a r i n g midway between the c e n t r e and t h e edge the s t r e s s has s u b s t a n t i a l l y i n c r e a s e d . The symmetry i s a l m o s t c i r c u l a r e x c e p t at the edge where the e i g h t - f o l d symmetry i s s t r o n g e r t h a n i n ( a ) . In ( c ) t h e s t r e s s a t t h e p e r i p h e r y has d e c r e a s e d 30 * w i t h r e s p e c t t o ( b ) . At the c e n t r e t h e s t r e s s has d o u b l e d . In the c e n t r a l a r e a t h e symmetry has t a k e n a more d e f i n i t e f o u r - f o l d p a t t e r n w h i c h a p p e a r s a g a i n i n t h e r i n g midway between the c e n t r e and t h e edge. In t h i s c a s e t h e r e i s a s l i g h t t w o - f o l d symmetry which can be seen as an e l o n g a t i o n of t h e minimum s t r e s s a r e a i n t h e [110] d i r e c t i o n w i t h 179 a 180 181 182 g u r e 9.25 MRSS-CRSS ( Y i e l d ) (MPa) i n a (001) p l a n e at a d i s t a n c e of 8.25 mm from the cone a t f o u r c r y s t a l l e n g t h s . (a) 13.75 mm ; (b) 27.5 mm ; ( c ) 55.0 mm ; (d) 82.5 mm. C r y s t a l r a d i u s 27.5 mm. 183 r e s p e c t t o the [110] d i r e c t i o n . In (d) t h e s t r e s s l e v e l has d e c r e a s e d i n t h e whole wafer and t h e r e f o r e no more d i s l o c a t i o n s a r e g e n e r a t e d w i t h f u r t h e r g r o w t h . At t h e end o f growth the f i n a l d i s l o c a t i o n d e n s i t y d i s t r i b u t i o n i n t h i s w a f e r w i l l be d e t e r m i n e d by the maximum s t r e s s r e a c h e d d u r i n g g r o w t h . At the edge t h i s s t r e s s i s r e a c h e d i n ( b ) . At t h e c e n t r e and midway r e g i o n t h e l a r g e s t s t r e s s i s r e a c h e d i n ( c ) . From t h e s e maximum v a l u e s i t may be p r e d i c t e d t h a t the w a f e r w i l l show a W-shaped d i s l o c a t i o n d i s t r i b u t i o n a l o n g a d i a m e t e r . The b r a n c h e s of the W w i l l be more t h a n t w i c e as h i g h as the c e n t r e . The e i g h t - f o l d symmetry a t t h e edge w i l l be h a r d l y n o t i c e d s i n c e i t o c c u r s i n a narrow r i n g w i t h d i s l o c a t i o n v a r i a t i o n s of t h e o r d e r o f 10 % . The f o u r - f o l d symmetry w i l l be a l s o v e r y weak. 2) S e c t i o n at 49.5 mm from the t h e cone. CL : (a) 55.0 mm; (b) 82.5 mm . The s t r e s s d i s t r i b u t i o n s f o r c r o s s - s e c t i o n s at 49.5 mm from the cone i s shown i n F i g u r e 9.26(a) and ( b ) . F i g u r e 9.26(a) c o r r e s p o n d s to a s t r e s s d i s t r i b u t i o n c l o s e t o the i n t e r f a c e . The f o l l o w i n g c h a r a c t e r i s t i c s can be d i s t i n g u i s h e d . The c e n t r a l p a r t has f o u r - f o l d symmetry and t h e r e s t o f t h e s u r f a c e has a l m o s t c i r c u l a r symmetry. The s t r e s s l e v e l i s t h e same a t t h e c e n t r e and t h e edge, and t h e minimum o c c u r s c l o s e t o the edge. In (b) t h e s t r e s s l e v e l i n t h e c e n t r a l r e g i o n has i n c r e a s e d by 50 % and a t the edge the s t r e s s has d o u b l e d w i t h r e s p e c t to ( a ) . The symmetry i n (b) i s a l m o s t c i r c u l a r t h r o u g h o u t the 184 185 F i g u r e 9.26 MRSS-CRSS ( Y i e l d ) (MPa) i n a (001) p l a n e a t a d i s t a n c e of 49.5 mm from t h e cone a t two c r y s t a l l e n g t h s . (a) 55.0 mm ; (b) 82.5 mm. C r y s t a l r a d i u s 27.5 mm. 186 s e c t i o n . The f i n a l maximum d i s l o c a t i o n d e n s i t y i n t h i s w a f e r may be d e t e r m i n e d by t h e s t r e s s e s d e v e l o p e d i n ( b ) . In t h i s c a s e a W- shaped d i s t r i b u t i o n i s p r e d i c t e d w i t h t h e c e n t e r o f the W o f the same h e i g h t as t h e b r a n c h e s . S i n c e a c o n s i d e r a b l e number of d i s l o c a t i o n s a r e g e n e r a t e d i n (a) w i t h f o u r - f o l d symmetry, t h i s w i l l be p r e s e n t i n t h e f i n a l d i s t r i b u t i o n . When t h e d i s l o c a t i o n d e n s i t y d i s t r i b u t i o n p r e d i c t e d i n t h i s s e c t i o n i s compared w i t h t h e f i n a l d i s l o c a t i o n d i s t r i b u t i o n g i v e n i n (1) f o r a s e c t i o n c l o s e t o the cone, the p r e d i c t e d d i s l o c a t i o n s f a r from the cone w i l l be 10 % l o w e r i n t h e o u t e r r e g i o n and 50 % h i g h e r i n t h e c e n t r e . 3) S e c t i o n at 77.0 mm from t h e cone. CL (a) 82.5 mm; (b) 110.0 mm . The s t r e s s d i s t r i b u t i o n s f o r a waf e r 77 mm from the cone a r e shown i n F i g u r e 9.27(a) and ( b ) . S i m i l a r p a t t e r n s as i n 2) a r e o b s e r v e d i n t h i s c a s e . The d i f f e r e n c e s a r e t h a t i n (a) the f o u r - f o l d symmetry i s expanded and i n (b) t h e r e a r e f o u r minimum s t r e s s r e g i o n s midway between the c e n t r e and the edge i n the <100> d i r e c t i o n s . 4) S e c t i o n a t 104.5 mm from the cone. CL : 110.0 mm. cone and The o n l y c r o s s - c o r r e s p o n d s to i t i s c l o s e t o s e c t i o n a v a i l a b l e a t t h i s d i s t a n c e from the the l o n g e s t c r y s t a l shown i n F i g u r e 9.21(e) t h e s o l i d / l i q u i d i n t e r f a c e . The s t r e s s 187 a 188 F i g u r e 9.27 MRSS-CRSS ( Y i e l d ) (MPa) i n a (001) p l a n e a t a d i s t a n c e of 77.0 mm from t h e cone at two c r y s t a l l e n g t h s , (a) 82.5 mm ; (b) 110.0 mm. C r y s t a l r a d i u s 27.5 mm. 189 d i s t r i b u t i o n i n t h i s s e c t i o n i s v e r y s i m i l a r t o the d i s t r i b u t i o n shown i n F i g u r e 9 . 2 7 ( a ) . I f growth i s c o m p l e t e d when t h i s l e n g t h i s r e a c h e d t h e d i s l o c a t i o n d i s t r i b u t i o n a t t h e t a i l end w i l l have th e f e a t u r e s o b s e r v e d i n F i g u r e 9 . 2 7 ( a ) . T h e s e a r e 1) an expanded f o u r - f o l d symmetry, 2) a d i s l o c a t i o n d i s t r i b u t i o n i n t h e r a d i a l d i r e c t i o n w i t h a W shape a c r o s s the d i a m e t e r h a v i n g a minimum v e r y c l o s e to t h e edge, a wide peak i n t h e c e n t r e , and b r a n c h e s e q u a l l y h i g h . The a v e r a g e d i s l o c a t i o n d e n s i t y i n t h i s w a f e r w i l l be h a l f of t h e d i s l o c a t i o n d e n s i t y f o r a wafer c o n s i d e r e d i n 3) a t 27.5 mm. I f i n s t e a d of f o l l o w i n g t h e s t r e s s d i s t r i b u t i o n s d u r i n g g r o w t h i n s p e c i f i c s e c t i o n s , the d i s t r i b u t i o n s of s t r e s s e s a r e o b s e r v e d a l o n g the c r y s t a l a t e a c h o f t h e f i v e l e n g t h s c o n s i d e r e d , t h e f o l l o w i n g p a t t e r n s can be o b s e r v e d . 1) The e i g h t - f o l d symmetry a t t h e edge o f the s e c t i o n s i s al w a y s p r e s e n t . T h i s symmetry i s a s s o c i a t e d w i t h an e i g h t - f o l d d i s t r i b u t i o n of t h e s l i p mode of the MRSS. 2) In t h e r a d i a l d i r e c t i o n , t h e s t r e s s d i s t r i b u t i o n a lways shows a W shape. 3) The f o u r - f o l d symmetry i s s t r o n g e r c l o s e t o t h e i n t e r f a c e between 5.5 mm and 11.0 mm. I t a l s o a p p e a r s at t h e top of t h e c r y s t a l n e a r t h e cone. 190 4) A s l i g h t t e n d e n c y to a t w o - f o l d symmetry when p r e s e n t i s a l w a y s a c c o m p a n i e d by a f o u r - f o l d symmetry. 5) F a r from t h e ends, at l e a s t 13.75 mm, t h e s t r e s s symmetry i s a l m o s t a x i s y m m e t r i c w i t h s l i g h t f o u r - f o l d symmetry a t the c e n t e r and e i g h t - f o l d symmetry a t the edge . 6) F o r c r y s t a l l e n g t h s l a r g e r t h a n 55.0 mm, c r o s s - s e c t i o n s f a r from t h e i n t e r f a c e have the minimum s t r e s s l e v e l s i n t h e <100> d i r e c t i o n s r a t h e r t h a n i n t h e <110> d i r e c t i o n s . The a r e a where t h i s minimum o c c u r s i s v e r y smal1 . 9.3 C r y s t a l R a d i u s C o n d i t i o n s : R, 40.0 mm ; CA, 30° ; B, 21.0 mm ; AP, 30 atm ; CL (a) 10.0 mm ; (b) 40.0 mm ; (c) 80.0 mm ; (d) 100.0 mm . The e f f e c t of i n c r e a s i n g t h e c r y s t a l r a d i u s from 27.5 mm t o 40 mm on i n t e r n a l c r y s t a l s t r e s s e s and d i s l o c a t i o n d i s t r i b u t i o n s i s d e t e r m i n e d a s s u m i n g t h a t t h e t e m p e r a t u r e p r o f i l e i n t h e medium s u r r o u n d i n g the c r y s t a l has not changed when t h e r a d i u s i s i n c r e a s e d . In t h i s c a s e t h e t e m p e r a t u r e p r o f i l e i s g i v e n i n F i g u r e 8.19 ( c u r v e 4 ) . V a l u e s f o r t h e MRSS at t h e f o u r c r y s t a l l e n g t h s c o n s i d e r e d a r e shown i n F i g u r e 9 . 2 8 ( a - d ) . The MRSS d i s t r i b u t i o n s f o l l o w the 191 same g e n e r a l p a t t e r n s as t h o s e o b s e r v e d w i t h the s m a l l e r c r y s t a l r a d i u s , w i t h two main d i f f e r e n c e s . The f i r s t i s t h a t the maximum s t r e s s , c o n s i d e r i n g t h e whole growth p r o c e s s , o c c u r s when t h e c r y s t a l i s 40.0 mm l o n g i n ( b ) , w i t h an a s p e c t r a t i o of 1 r a t h e r t h a n 2 as i n the s m a l l e r c r y s t a l . T h i s d i f f e r e n c e i s c o n s i s t e n t w i t h the e x p l a n a t i o n g i v e n i n t h e l a s t s e c t i o n . The maximum s t r e s s u s u a l l y o c c u r s above t h e e n c a p s u l a n t s u r f a c e . I f t h i s p o s i t i o n c o i n c i d e s w i t h t h e m i d l e n g t h p o s i t i o n i n the c r y s t a l t h e r e i s an e x t r a c o n t r i b u t i o n to t h e s t r e s s due t o the q u a s i - p l a n e s t r a i n c o n d i t i o n s p r e v a i l i n g t h e r e . The s e c o n d d i f f e r e n c e i s t h e h i g h e r s t r e s s l e v e l f o r t h e l a r g e r c r y s t a l . In t h i s c a s e i n c r e a s i n g the r a d i u s 45 % i n c r e a s e s t h e s t r e s s e s by 40 % i n d i c a t i n g t h a t the MRSS i n c r e a s e s p r o p o r t i o n a l l y w i t h the r a d i u s . The e f f e c t of i n c r e a s i n g t h e r a d i u s on s t r e s s l e v e l s has been e x p l a i n e d i n t h e l i t e r a t u r e by c o n s i d e r i n g the e f f e c t of i n c r e a s i n g the r a d i u s on t h e t h e r m a l f i e l d . As p o i n t e d out p r e v i o u s l y , the r e s u l t s f o r t h e e f f e c t o f r a d i u s on t e m p e r a t u r e f i e l d s a r e c o n t r a d i c t o r y . T h i s c o u l d r e s u l t from t h e l i m i t a t i o n s o f the s o l u t i o n s employed t o s t u d y t h i s e f f e c t . In the p r e s e n t i n v e s t i g a t i o n t h e e f f e c t of r a d i u s on t h e r m a l g r a d i e n t s i s f i r s t c o n s i d e r e d as a p o s s i b l e c a u s e f o r the o b s e r v e d e f f e c t on s t r e s s e s . The c o m p a r i s o n can be made u s i n g two c r i t e r i a . I t can be made f o r two c r y s t a l s of same a s p e c t r a t i o , or between two c r y s t a l s w i t h t h e same l e n g t h . In t h i s i n v e s t i g a t i o n the s e c o n d method i s c h o s e n . (The c o n c l u s i o n s o b t a i n e d w i t h b o t h methods a r e s i m i l a r ) . F o r t h e c o m p a r i s o n c r y s t a l l e n g t h s f o r which the 192 maximum s t r e s s e s o c c u r a r e c h o s e n . F o r t h e 27.5 mm r a d i u s c r y s t a l t h e l e n g t h i s 55 mm. F o r the 40.0 mm r a d i u s c r y s t a l t h e l e n g t h i s 40 mm. Because the l e n g t h s a r e d i f f e r e n t , a n o t h e r l e n g t h f o r the 40 mm r a d i u s c r y s t a l i s a l s o c o n s i d e r e d , s p e c i f i c a l l y 80 mm l e n g t h . F o r the t h r e e c r y s t a l s , g r a d i e n t s i n two d i r e c t i o n s a r e c o n s i d e r e d ; a x i a l g r a d i e n t s a l o n g t h e c r y s t a l a x i s and a v e r a g e r a d i a l g r a d i e n t . The c a l c u l a t e d g r a d i e n t s a r e shown i n F i g u r e 9.29(a) f o r the a x i a l g r a d i e n t s and (b) f o r t h e r a d i a l g r a d i e n t s as a f u n c t i o n of d i s t a n c e from the i n t e r f a c e . Curve (1) c o r r e s p o n d s t o t h e 27.5 mm c r y s t a l and c u r v e s (2) and (3) f o r t h e 40 mm c r y s t a l . I t can be o b s e r v e d i n the f i g u r e s t h a t the c u r v e (3) i s below c u r v e (1) i n (a) and c u r v e (1) i s between (2) and (3) i n ( b ) . From t h e s e o b s e r v a t i o n s i t can be c o n c l u d e d t h a t the t h e r m a l f i e l d i s not i n f l u e n c e d by changes i n r a d i u s and t h a t the l a r g e change i n s t r e s s o b s e r v e d c a n n o t be a c c o u n t e d f o r by the change i n t h e r m a l f i e l d . An a l t e r n a t i v e e x p l a n a t i o n f o r t h e e f f e c t of r a d i u s on s t r e s s i s r e l a t e d to the s p e c i f i c g e o m e t r y of t h e c r y s t a l . In the c a s e of c y l i n d r i c a l b o d i e s t h e r a d i a l g r a d i e n t has a d i f f e r e n t e f f e c t on s t r e s s e s t han i n c u b i c g e o m e t r i e s . In c y l i n d e r s a c o n s t a n t r a d i a l g r a d i e n t can d e v e l o p s t r e s s e s ; i n cubes i t c a n n o t . A r a d i a l g r a d i e n t i n a c y l i n d e r g e n e r a t e s i n c o m p a t i b i l i t i e s i n the s t r a i n components f o r f r e e e x p a n s i o n . From t h e p r e s e n t r e s u l t s i t i s c o n c l u d e d t h a t the same r a d i a l g r a d i e n t i n two c y l i n d e r s of d i f f e r e n t r a d i u s c r e a t e s l a r g e r i n c o m p a t i b i l i t i e s i n a c r y s t a l w i t h l a r g e r r a d i u s . T h i s i s more c l e a r l y s een when i t i s c o n s i d e r e d t h a t t h e e f f e c t of t h e r m a l  194 F i g u r e 9.29 (a) A x i a l a l o n g the c r y s t a l a x i s and (b) r a d i a l t h e r m a l g r a d i e n t s as a f u n c t i o n of d i s t a n c e from th e i n t e r f a c e f o r two c r y s t a l r a d i u s . Curve (1) f o r a r a d i u s of 27.5 mm ; and c u r v e s (2) and (3) f o r a r a d i u s of 40.0 mm. C r y s t a l l e n g t h s a r e (1) 55.0 mm ; (2) 40.0 mm and (3) 80.0 mm. 195 s t r a i n i s i n c l u d e d as an i n i t i a l s t r a i n c a l c u l a t e d as a f r e e e x p a n s i o n f r o m a r e f e r e n c e t e m p e r a t u r e . T h i s f r e e e x p a n s i o n ( c o n t r a c t i o n i n t h i s c a s e ) i n c r e a s e s w i t h i n c r e a s i n g d i s t a n c e f r o m t h e c e n t r e . The d e g r e e o f i n c o m p a t i b i l i t y i s e x p e c t e d t o f o l l o w t h e e x p a n s i o n , w h i c h i n t u r n w i l l g e n e r a t e l a r g e r s t r e s s e s . The e f f e c t o f a c o n s t a n t r a d i a l g r a d i e n t G on t h e s t r e s s i n c y l i n d e r s o f d i f f e r e n t r a d i i c an be q u a n t i t a t i v e l y e v a l u a t e d u s i n g t h e p l a n e s t r a i n a p p r o x i m a t i o n . The s t r e s s components f o r a t e m p e r a t u r e f i e l d T = -Gr G a E Or - ( r - r ) 3 1 - V <2r - r 0 ) a E 1 - v Oz = G ( r - 2 / 3 r Q ) a E l - v I t c a n be s e e n t h a t t h e s t r e s s c o m p o n e n t s i n c r e a s e s l i n e a r l y w i t h t h e r a d i u s r . The MRSS-CRSS ( Y i e l d ) c o n t o u r s f o r t h e f o u r l e n g t h s c o n s i d e r e d a r e shown i n F i g u r e 9 . 3 0 ( a - d ) . The r e s u l t s shown a r e s i m i l a r t o t h o s e o b t a i n e d w i t h t h e s m a l l e r c r y s t a l , w i t h a r e a s h a v i n g s t r e s s l e v e l s b e l o w t h e CRSS i n ( a ) and ( d ) . The d i s t r i b u t i o n o f t h e MRSS-CRSS ( Y i e l d ) f o r t h e f o u r c r y s t a l  197 l e n g t h s show c o n t o u r s and l e v e l s w hich a r e v e r y s i m i l a r t o the MRSS c o n t o u r s . The d i s t r i b u t i o n of MRSS-CRSS (MBTe) a r e s i m i l a r to t h o s e i n F i g u r e 9 . 3 0 ( a - d ) . 9.3.1 E f f e c t of R a d i u s on S t r e s s Symmetry The s t r e s s symmetry i n c r y s t a l s w i t h f o u r d i f f e r e n t l e n g t h s has been examined. The c r o s s - s e c t i o n s s t u d i e d c o r r e s p o n d t o t h e same r e l a t i v e p o s i t i o n s c o n s i d e r e d i n the c r y s t a l s w i t h s m a l l e r r a d i u s . As b e f o r e , t h e a n a l y s i s was done i n two ways namely f o l l o w i n g a f i x e d c r o s s - s e c t i o n i n the c r y s t a l d u r i n g g r o w t h and a n a l y s i n g the symmetry d i s t r i b u t i o n i n t h e c r y s t a l f o r each c r y s t a l l e n g t h . From the s e c o n d a n a l y s i s , t h e c o n c l u s i o n s a r e s i m i l a r t o t h o s e o b t a i n e d f o r t h e s m a l l e r r a d i u s . From t h e f i r s t a n a l y s i s by f o l l o w i n g a c r o s s - s e c t i o n and c o m p a r i n g the r e s u l t s w i t h t h o s e o b t a i n e d f o r t h e s m a l l e r r a d i u s c r y s t a l , t h e f o l l o w i n g c o n c l u s i o n s were r e a c h e d . T h e r e i s a c o r r e l a t i o n of symmetry w i t h d i s t a n c e from th e cone between the two c r y s t a l r a d i i . T h i s c o r r e l a t i o n i s g i v e n i n T a b l e 9.4 which was o b t a i n e d i n t h e f o l l o w i n g way. C o n s i d e r t h e s e c t i o n s at 77 mm (2.8 x R) f o r the 27.5 mm c r y s t a l and 72 mm (1.8 x R) f o r the 40 mm c r y s t a l . B o th s e c t i o n s have v e r y s i m i l a r s t r e s s s y m m e t r i e s d u r i n g g r o w t h . In t h i s c a s e the s t r e s s symme- t r i e s shown i n F i g u r e 9.27(a) and (b) c o r r e s p o n d i n g t o a r e l a t i v e d i s t a n c e from the cone of 2.8 t i m e s the r a d i u s i s r e p r o d u c e d i n t h e l a r g e r c r y s t a l but a t a r e l a t i v e d i s t a n c e of 1.8 t i m e s th e r a d i u s . S i m i l a r c o r r e l a t i o n s a r e o b t a i n e d f o l l o w i n g o t h e r s e c t i o n at t h e same d i s t a n c e s from th e cones f o r the two r a d i u s . 198 T a b l e 9. 4 T a b l e of Symmetry C o r r e l a t i o n s R a d i u s [mm ] D i s t a n c e from R e l a t i v e D i s t a n c e | S h o u l d e r [mm] | from s h o u l d e r | F i g u r e 27 40 . 5 . 0 8.25 0.3 12.0 0.3 9 . 25 27 40 . 5 . 0 77.0 2.8 72.0 1.8 9 . 26 27 40 . 5 . 0 104.5 3.8 112.0 2.8 9 . 27 Because t h i s c o r r e l a t i o n i s w i t h a c t u a l d i s t a n c e , the e f f e c t of t h e t h e r m a l f i e l d i s e x p e c t e d t o p r e d o m i n a t e o v e r the e f f e c t of t h e boundary c o n d i t i o n s . I f t h e symmetry i s d e t e r m i n e d by the t h e r m a l f i e l d a l o n e , t h e r e s h o u l d a l s o be a s t r o n g c o r r e l a t i o n between the s y m m e t r i e s and t h e a c t u a l d i s t a n c e from the i n t e r f a c e . T h i s c o r r e l a t i o n i s not o b s e r v e d . On t h e c o n t r a r y , f o r l o n g c r y s t a l s t h e r e i s c o r r e l a t i o n o f s y m m e t r i e s but w i t h the r e l a t i v e d i s t a n c e from t h e i n t e r f a c e f o r same c r y s t a l l e n g t h s . The c o r r e l a t i o n w i t h a c t u a l d i s t a n c e from t h e cone, however, s u g g e s t s t h a t t h e t h e r m a l h i s t o r y of t h e w a f e r i s i m p o r t a n t i n d e t e r m i n i n g the s t r e s s symmetry. T h i s would r e s u l t s i n s e c t i o n s p a s s i n g t h r o u g h t h e same t h e r m a l f i e l d s h a v i n g t h e same symmetry h i s t o r y r e g a r d l e s s of t h e c r y s t a l r a d i u s . 199 9.4 Growth V e l o c i t y C o n d i t i o n s R , 27.5 mm CL, 55.0 mm B, 21 mm T e m p e r a t u r e p r o f i l e as i n F i g u r e 8.19, c u r v e 4, V e l o c i t y (a) 0.0001 cm/s, (b) 0.001 cm/s , (c) 0.01 cm/s To d e t e r m i n e t h e e f f e c t of changes i n t h e g r o w th v e l o c i t y on t h e t h e r m a l and s t r e s s f i e l d s , t h e c a l c u l a t i o n s were r e p e a t e d w i t h g r o w th v e l o c i t i e s of 0.0001 and 0.01 cm/s. The t e m p e r a t u r e f i e l d s f o r t h e t h r e e v e l o c i t i e s c o n s i d e r e d a r e shown i n F i g u r e 9 . 3 1 ( a - c ) . C o m paring F i g u r e 9.31(b) and ( a ) , i t i s n o t e d t h a t d e c r e a s i n g t h e v e l o c i t y has l i t t l e e f f e c t on the t e m p e r a t u r e d i s t r i b u t i o n . I n c r e a s i n g t h e v e l o c i t y ( F i g u r e 9.31(b) and (c) ) s i g n i f i c a n t l y i n c r e a s e s the a x i a l and r a d i a l t e m p e r a t u r e g r a d i e n t s i n t h e c r y s t a l . The r e s u l t s i n d i c a t e t h a t a t v e l o c i t i e s of 0.001 cm/s and l o w e r the t h e r m a l system i s a t s t e a d y s t a t e , and e f f e c t i v e l y i n d e p e n d e n t of v e l o c i t y . Above 0.001 cm/s t h i s i s not t h e c a s e , p r i m a r i l y due t o the v a l u e o f t h e r m a l d i f f u s i v i t y used (0.04 cm/s) w h i c h i s c o m p a r a b l e t o t h e v e l o c i t y . The s p e c i f i c e f f e c t of g r o w t h v e l o c i t y on the t h e r m a l f i e l d s i s complex. In t h e q u a s i s t e a d y s t a t e a p p r o x i m a t i o n , t h e t e m p e r a t u r e f i e l d c h anges o n l y as a r e s u l t of the movement of the s o l i d i f y i n g i n t e r f a c e ; t h e i n i t i a l t r a n s i e n t i s n e g l e c t e d . In t h e p r e s e n t c a l c u l a t i o n s , and i n d e p e n d e n t of the h e a t t r a n s f e r a t the c r y s t a l 200 201 C F i g u r e 9.31 T e m p e r a t u r e f i e l d s f o r t h r e e growth v e l o c i t i e s (a) 0.0001 cm/s, (b) 0.001 cm/s and ( c ) 0.01 cm/s. T e m p e r a t u r e g i v e n i n 10 °C. L i t t l e change i s o b s e r v e d from (a) to ( b ) . From (b) to ( c ) g r a d i e n t s have i n c r e a s e d . R a d i u s 27.5 mm, l e n g t h 55 mm, e n c a p s u l a n t t h i c k n e s s 21 mm. •-%f\L f'U 202 s u r f a c e , t h e r e a r e two l i m i t i n g c a s e s . At low v e l o c i t i e s s t e a d y s t a t e f i e l d s a r e e x p e c t e d ; a t h i g h v e l o c i t i e s u n i f o r m t e m p e r a t u r e f i e l d s a r e e x p e c t e d w i t h t h e t e m p e r a t u r e the same as th e moving b o u n d a r y . F o r c o m p a r a b l e v a l u e s of v e l o c i t y and d i f f u s i v i t y any q u a l i t a t i v e a n a l y s i s must i n c l u d e t h e f a c t t h a t t h e e x t e r n a l s u r f a c e i s c o o l i n g f o l l o w i n g Newton's law of c o o l i n g . In t h i s c a s e t h e s i t u a t i o n i s c o m p l i c a t e d by t h e f a c t t h a t t h e r e i s mixed c o n t r o l i n t h e b u l k and the s u r f a c e o f t h e c r y s t a l w i t h a B i o t number w h i c h has v a l u e s c l o s e to one. The MRSS c o n t o u r s c o r r e s p o n d i n g t o t h e t h r e e t h e r m a l f i e l d s shown i n F i g u r e 9.31(a-c) a r e g i v e n i n F i g u r e 9 . 3 2 ( a - c ) . C o m p aring t h e MRSS v a l u e s a t 0.0001 cm/s w i t h t h o s e a t 0.001 cm/s, t h e d i f f e r e n c e s a r e s m a l l ; a p p r o x i m a t e l y 2 % f o r the h i g h e s t v a l u e s of the MRSS and 4 % f o r t h e l o w e s t . Comparing 0.01 cm/s w i t h 0.001 cm/s, the h i g h e s t v a l u e s o f MRSS have a p p r o x i m a t e l y d o u b l e d and the l o w e s t v a l u e s more t h a n d o u b l e d . In a d d i t i o n , t h e a r e a c l o s e to the i n t e r f a c e w i t h minimum MRSS d i s a p p e a r s as t h e v e l o c i t y i n c r e a s e s . 9.5 T h e r m a l C o n d i t i o n s The t e m p e r a t u r e d i s t r i b u t i o n i n t h e g r o w i n g c r y s t a l i s depe n d e n t on t h e t h e r m a l e n v i r o n m e n t s u r r o u n d i n g t h e c r y s t a l and the c r y s t a l g e o m e t r y . The e f f e c t o f t h e t h e r m a l e n v i r o n m e n t i s m o d e l l e d u s i n g t h e t e m p e r a t u r e measurements r e p o r t e d by Gr a n t e t 218 a l . as shown i n F i g u r e 8.19, c u r v e 4.  204 C F i g u r e 9.32 MRSS-CRSS ( Y i e l d ) (MPa) f o r the t h r e e t e m p e r a t u r e f i e l d s shown i n F i g u r e 9.31(a-c) c o r r e s p o n d i n g t o t h r e e growth v e l o c i t i e s . (a) 0.0001 cm/s, (b) 0.001 cm/s and ( c ) 0.01 cm/s. In (a) and (b) t h e s t r e s s f i e l d s a r e s i m i l a r . In ( c ) t h e s t r e s s d i s t r i b u t i o n and s t r e s s v a l u e s change. 205 The t h e r m a l p r o f i l e f o r the d i f f e r e n t t h i c k n e s s e s of bo r o n o x i d e i s m o d e l l e d a s s u m i n g t h a t the t e m p e r a t u r e at the top and bott o m o f the o x i d e i s i n d e p e n d e n t o f t h i c k n e s s f o r c o n s t a n t power i n p u t . T h i s makes the t e m p e r a t u r e g r a d i e n t a c r o s s the o x i d e l i n e a r l y p r o p o r t i o n a l to the t h i c k n e s s , as d e m o n s t r a t e d 14 6-148 e x p e r i m e n t a l l y I t i s a l s o assumed t h a t r e d u c i n g t h e a x i a l g r a d i e n t s i n t h e chamber by i n c r e a s i n g the power i n p u t or i n c r e a s i n g the b o r o n o x i d e t h i c k n e s s changes the magn i t u d e o f t h e l o c a l t e m p e r a t u r e but does not change the shape o f the t e m p e r a t u r e p r o f i l e . T h i s means t h a t o n l y the c o e f f i c i e n t s i n t h e p o l y n o m i a l s f i t t i n g t h e o r i g i n a l p r o f i l e a r e p r o p o r t i o n a l l y r e d u c e d by a g i v e n f a c t o r f o r th e new growth c o n d i t i o n s . The b o r o n o x i d e t h i c k n e s s e s c o n s i d e r e d a r e 21, 40 and 50 mm the l a t t e r b e i n g t h e maximum t h i c k n e s s c u r r e n t l y used e x p e r i m e n t a l l y . F o r each t h i c k n e s s , s e v e r a l t e m p e r a t u r e g r a d i e n t s a r e c o n s i d e r e d . The c r y s t a l r a d i i examined a r e 27.5 mm and 40.0 mm. 9.5.1 R a d i u s 27.5 mm 9.5.1.1 Boron o x i d e t h i c k n e s s 21.0 mm A v e r a g e g r a d i e n t s (a) 66°c/cm, (b) 33°C/cm, (c) 17°C/cm, (d) 8°C/cm The t e m p e r a t u r e p r o f i l e s c o n s i d e r e d a r e g i v e n i n F i g u r e 9.33. F i v e p r o f i l e s a r e shown. The p r o f i l e l a b e l l e d G w i t h t h e 206 F i g u r e 9.33 T e m p e r a t u r e p r o f i l e s a l o n g t h e c r y s t a l s u r f a c e i n the e n v i r o n m e n t s u r r o u n d i n g t h e c r y s t a l employed i n the c a l c u l a t i o n s . The p r o f i l e s a r e d e r i v e d from F i g u r e 8.19, c u r v e 4, r e p r o d u c e d as c u r v e G. The v a l u e s of g r a d i e n t a r e a v e r a g e d i n the c r y s t a l l e n g t h c o n s i d e r e d . E n c a p s u l a n t t h i c k n e s s 21 mm. 207 s t e e p e s t g r a d i e n t , i s t h e same as shown i n F i g u r e 8.19, c u r v e 4, w h i c h was d e t e r m i n e d e x p e r i m e n t a l l y . The r e s u l t s f o r t h e MRSS-CRSS ( Y i e l d ) f o r f o u r t e m p e r a t u r e p r o f i l e s a r e shown i n F i g u r e 9 . 3 4 ( a - d ) . The s t r e s s c o n t o u r s f o r t h e t e m p e r a t u r e p r o f i l e G were shown i n F i g u r e 9 . 3 3 ( c ) . C o m paring F i g u r e 9.34(a-d) i t can be n o t e d t h a t t h e s t r e s s d i s t r i b u t i o n p a t t e r n s a r e s i m i l a r i n t h e f o u r c a s e s . The s t r e s s l e v e l s , however, d e c r e a s e s c o n s i d e r a b l y w i t h t h e a v e r a g e g r a d i e n t . As a r e s u l t of the lower s t r e s s l e v e l s the a r e a w i t h z e r o s t r e s s g r a d u a l l y i n c r e a s e s from (a) t o ( d ) . The r e g i o n s w i t h z e r o s t r e s s s t a r t to grow from the r e g i o n s w i t h lower s t r e s s e s . These r e g i o n s a r e i n t h e s e e d , s h o u l d e r and above the i n t e r f a c e edge. At t h e l o w e s t g r a d i e n t i n (d) most o f t h e c r y s t a l has s t r e s s e s w h i c h a r e l e s s t h a n th e CRSS ( Y i e l d ) . The two r e m a i n i n g r e g i o n s w i t h non z e r o s t r e s s e s a r e l o c a t e d where the h i g h e s t s t r e s s e s u s u a l l y a p p e a r ; i n t h e c r y s t a l s u r f a c e above the e n c a p s u l a n t and i n the c r y s t a l a x i s . 9.5.1.2 Boron o x i d e t h i c k n e s s 40.0 mm AVG (a) 58 cm/s, (b) 29 cm/s, ( c ) 19 cm/s B 50.0 mm AVG (a) 50°c/cm. (b) 25°C/cm, ( c ) 12.5°C/cm.  209 c d F i g u r e 9.34 MRSS-CRSS ( Y i e l d ) (MPa) f o r f o u r t e m p e r a t u r e p r o f i l e s w i t h a v e r a g e g r a d i e n t s (a) 33 C/cm, (b) 17 C/cm, (c) 11 C/cm and (d) 8 C/cm. R a d i u s 27.5 mm, l e n g t h 55 mm. E n c a p s u l a n t t h i c k n e s s 21 mm. 210 The t e m p e r a t u r e p r o f i l e s f o r b o t h e n c a p s u l a n t t h i c k n e s s e s a r e shown i n F i g u r e s 9.35(a) and ( b ) . In (a) f o r t h e 40 mm e n c a p s u l a n t and i n (b) f o r the 50 mm. In e a c h c a s e t h r e e t e m p e r a t u r e p r o f i l e s a r e c o n s i d e r e d , t h e a v e r a g e g r a d i e n t f o r each p r o f i l e i s a l s o g i v e n . F o r c o m p a r i s o n t h e o r i g i n a l p r o f i l e f o r the 21.0 mm e n c a p s u l a n t i s a l s o shown. The r e s u l t s f o r the MRSS-CRSS ( Y i e l d ) f o r the two e n c a p s u l a n t t h i c k n e s s e s a r e shown i n F i g u r e s 9 . 3 6 ( a - c ) f o r t h e 40 mm t h i c k n e s s and F i g u r e 9.37(a-c) f o r t h e 50 mm t h i c k n e s s . P a r t s (a) t o ( c ) c o r r e s p o n d s to t h e t h r e e d i f f e r e n t t h e r m a l p r o f i l e s shown i n F i g u r e 9.35(a) and ( b ) . To o b s e r v e t h e e f f e c t of i n c r e a s i n g t h e e n c a p s u l a n t t h i c k n e s s o n l y t h e t h r e e f i g u r e s to be compared a r e : 9 . 23(c) f o r t h e 21 mm, 9.36(a) f o r t h e 40 mm and 9.37(a) f o r t h e 50 mm t h i c k n e s s . I t can be n o t e d t h a t i n c r e a s i n g t h e b o r o n o x i d e t h i c k n e s s changes t h e r e l a t i v e s t r e s s d i s t r i b u t i o n . The changes a r e the e x t e n d e d low s t r e s s r e g i o n c l o s e to t h e i n t e r f a c e and the s h i f t of t h e s t r e s s c o n c e n t r a t i o n t o a p o s i t i o n above the e n c a p s u l a n t s u r f a c e . F o r t h i c k encapsu1 a n t s , the l a r g e and u n i f o r m low s t r e s s r e g i o n i n the low p a r t of the c r y s t a l has a s t r e s s d i s t r i b u t i o n i n a r a d i a l d i r e c t i o n w h i c h m a i n t a i n s t h e W shape w i t h e x t e n d e d v a l l e y s and s h a r p peaks i n the c e n t r e and edge o f t h e c r y s t a l . In t h e c a s e of t h e 50 mm t h i c k n e s s i n F i g u r e 9.37(a) t h e s t r e s s c o n c e n t r a t i o n r e g i o n below the s h o u l d e r i s s m a l l b e c a u s e the e n c a p s u l a n t i s o n l y 5 mm s h o r t e r t h a n th e c r y s t a l l e n g t h . 211 F i g u r e 9.35 T e m p e r a t u r e p r o f i l e s used i n t h e c a l c u l a t i o n s f o r two b o r o n o x i d e t h i c k n e s s e s . (a) 40 mm, (b) 50 mm. The p r o f i l e s a r e d e r i v e d from c u r v e G. The v a l u e s of g r a d i e n t s g i v e n a r e a v e r a g e d a l o n g t h e c r y s t a l l e n g t h . 211 F i g u r e 9.35 T e m p e r a t u r e p r o f i l e s used i n the c a l c u l a t i o n s f o r two b o r o n o x i d e t h i c k n e s s e s . (a) 40 mm, (b) 50 mm. The p r o f i l e s a r e d e r i v e d from c u r v e G. The v a l u e s of g r a d i e n t s g i v e n a r e a v e r a g e d a l o n g t h e c r y s t a l l e n g t h . 212 213 F i g u r e 9.36 MRSS-CRSS ( Y i e l d ) (MPa) f o r t h r e e a v e r a g e g r a d i e n t s . (a) 58°C/cm, (b) 29°C/cm, ( c ) 19°C/cm. R a d i u s 27.5 mm, l e n g t h 55 mm, b o r o n o x i d e t h i c k n e s s 40 mm. 214 215 F i g u r e 9.37 MRSS-CRSS ( Y i e l d ) (MPa) f o r t h r e e a v e r a g e g r a d i e n t s . (a) 50°C/cm, (b) 25 C/cm, ( c ) 17°C/cm. R a d i u s 27.5 mm, l e n g t h 55 mm, b o r o n o x i d e t h i c k n e s s 50 mm. 216 The a b s o l u t e s t r e s s a l s o changes w i t h an i n c r e a s e i n the e n c a p s u l a n t t h i c k n e s s . D o u b l i n g t h e t h i c k n e s s r e d u c e s by h a l f the maximum s t r e s s . When t h e r m a l p r o f i l e s w i t h l o w e r g r a d i e n t s a r e c o n s i d e r e d f o r b o t h e n c a p s u l a n t t h i c k n e s s e s , t h e r e g i o n i n t h e c r y s t a l w i t h z e r o MRSS-CRSS ( Y i e l d ) expands from t h e low s t r e s s r e g i o n s c l o s e to t h e i n t e r f a c e as the g r a d i e n t d e c r e a s e s . The l a r g e s t z e r o s t r e s s r e g i o n i s o b t a i n e d i n F i g u r e 9 . 37(c) f o r an e n c a p s u l a n t t h i c k n e s s of 50 mm and a g r a d i e n t o f 12.5°c/cm. I t i s n o t e d t h a t s i m i l a r s u r f a c e a r e a s were o b t a i n e d w i t h a 21 mm t h i c k n e s s f o r a g r a d i e n t o f 8°C/cm. T h i s shows t h a t i n d e p e n d e n t of e n c a p s u l a n t t h i c k n e s s , a c r i t i c a l g r a d i e n t of about 10°C/cm s h o u l d be a c h i e v e d i n o r d e r to o b t a i n l a r g e a r e a s w i t h s t r e s s e s l e s s t h a n the CRSS ( Y i e l d ) . From t h e p r a c t i c a l p o i n t o f view, a t h i c k e r e n c a p s u l a n t i s p r e f e r r e d f o r the f o l l o w i n g r e a s o n s . To g e t the d e s i r e d 10°C/cm l e v e l , t h e t h e r m a l g r a d i e n t s h o u l d be r e d u c e d e i g h t t i m e s w i t h the 21 mm t h i c k n e s s and o n l y t h r e e t i m e s w i t h t h e 50 mm t h i c k n e s s . T h i s i s b e c a u s e o f t h e r e d u c t i o n of g r a d i e n t s when i n c r e a s i n g the t h i c k n e s s . In a d d i t i o n i n low g r a d i e n t e n v i r o n m e n t s l a r g e a r e a s of c r y s t a l a r e e x p o s e d to h i g h t e m p e r a t u r e s . W i t h o u t p r o p e r e n c a p s u l a t i o n t h e r e i s d e c o m p o s i t i o n of GaAs, r e s u l t i n g i n t h e f o r m a t i o n of l i q u i d Ga dr o p s w h i c h damages t h e c r y s t a l and l e a d s to n o n - s t o i c h i o m e t r i c m e l t s w h i c h i s a l s o d e t r i m e n t a l . 217 9.5.2 R a d i u s 40.0 mm 9.5.2.1 Boron o x i d e t h i c k n e s s 21 mm AVG : a) 14°C/cm, (b) 7°C/cm The r e s u l t s f o r the MRSS-CRSS ( Y i e l d ) w i t h a b o r o n o x i d e l a y e r t h i c k n e s s o f 21 mm a r e shown i n F i g u r e 9.38(a) and ( b ) . P a r t (a) c o r r e s p o n d s t o t h e p r o f i l e l a b e l l e d G/4 ( 1 4 ° C / c m ) and (b) t o t h e p r o f i l e G/8 (7°C/cm) i n F i g u r e 9.33. I t i s n o t e d t h a t the a v e r a g e g r a d i e n t s i n t h e s e p r o f i l e s a r e l o w e r b e c a u s e of the l o n g e r c r y s t a l c o n s i d e r e d . C o m p aring F i g u r e 9.38(a) and (b) w i t h F i g u r e 9.34(c) and (d) f o r t h e 27.5 mm c r y s t a l , i t can be n o t e d t h a t i n t h e c a s e of t h e 40 mm c r y s t a l , t h e r e l a t i v e z e r o MRSS- CRSS ( Y i e l d ) a r e a f o r the same p r o f i l e i s not as l a r g e as i n t h e c a s e o f t h e 27.5 mm c r y s t a l . In t h e 40 mm c r y s t a l , however, t h e t o p of t h e c r y s t a l i n F i g u r e 9.38(b) shows z e r o s t r e s s . T h i s i s due t o t h e d i f f e r e n c e i n l e n g t h between the two c r y s t a l s . 9.5.2.2 Boron o x i d e t h i c k n e s s 40 mm AVG : (a) 46°C/cm, (b) 26°C/cm, (c) 15°C/cm, (d) 9°C/cm. The MRSS-CRSS ( Y i e l d ) c o n t o u r s f o r t h e f o u r g r a d i e n t s a r e shown i n F i g u r e 9 . 3 9 ( a - d ) . Note t h a t t h e p r o f i l e w i t h an a v e r a g e g r a d i e n t o f 9°C/cm i s not shown i n F i g u r e 9 . 3 5 ( a ) . The 9°C/cm a v e r a g e g r a d i e n t i n the e n c a p s u l a n t and gas i s f i v e t i m e s s m a l l e r t h a n t h e o r i g i n a l t e m p e r a t u r e p r o f i l e l a b e l l e d G i n F i g u r e 9.33. Comparing the r e s u l t s o f F i g u r e s 9.39(a-d) w i t h t h o s e i n F i g u r e 9.36(a-d) f o r t h e 27.5 mm c r y s t a l i t can be o b s e r v e d 218 F i g u r e 9.38 MRSS-CRSS ( Y i e l d ) (MPa) f o r two (a) 14°C/cm, (b) 7 C/cm. R a d i u s b o r o n o x i d e t h i c k n e s s 21 mm. a v e r a g e g r a d i e n t s . 40 mm, l e n g t h 80 mm,  220 c d F i g u r e 9.39 MRSS-CRSS ( Y i e l d ) (MPa) f o r f o u r a v e r a g e g r a d i e n t s . (a) 46°C/cm, (b) 26°C/cm, ( c ) 15°C/cm, (d) 9°C/cm. R a d i u s 40 mm, l e n g t h 80 mm, b o r o n o x i d e t h i c k n e s s 40 mm . 221 t h a t the c o n c e n t r a t i o n of s t r e s s e s i n the l a r g e r c r y s t a l o c c u r s a t t h e m i d l e n g t h i n s t e a d of t h e t o p of the c r y s t a l as i n t h e 27.5 mm c r y s t a l . In a d d i t i o n i t i s o b s e r v e d t h a t t o o b t a i n s i m i l a r r e l a t i v e a r e a s w i t h z e r o MRSS-CRSS ( Y i e l d ) , the g r a d i e n t s i n the e n v i r o n m e n t s u r r o u n d i n g t h e 40 mm c r y s t a l s h o u l d be f u r t h e r r e d u c e d . 9.5.2.3 Boron O x i d e t h i c k n e s s 50 mm, AVG : (a) 43°C/cm, (b) 14°C/cm, (c) 7°C/cm. In o r d e r t o d e t e r m i n e i f t h e s t r e s s l e v e l has r e a c h e d a maximum v a l u e c o n s i d e r i n g t h e whole growth p e r i o d , s t r e s s c a l c u l a t i o n s a r e made f o r a 40 mm c r y s t a l w i t h a l e n g t h c o m p a r a b l e t o t h e e n c a p s u l a n t t h i c k n e s s . The c r y s t a l l e n g t h i s 52 mm and t h e a v e r a g e g r a d i e n t i s 50°C/cm. The r e s u l t s f o r MRSS- CRSS ( Y i e l d ) a r e shown i n F i g u r e 9.40. I t i s o b s e r v e d t h a t the s t r e s s c o n c e n t r a t i o n o c c u r s below t h e s h o u l d e r . In F i g u r e 9.41(a) the d i s t r i b u t i o n f o r the same s t r e s s f o r a c r y s t a l l e n g t h of 80 mm i s shown. In t h i s c a s e t h e s t r e s s c o n c e n t r a t i o n o c c u r s above t h e e n c a p s u l a n t s u r f a c e . The s t r e s s v a l u e i n t h e l o n g e r c r y s t a l i s h i g h e r t h a n t h e l a r g e s t s t r e s s v a l u e i n t h e s h o r t e r c r y s t a l . From t h e s e r e s u l t s and t h o s e o b t a i n e d w i t h t h e 27.5 mm c r y s t a l r a d i u s i t i s p o s s i b l e t o g e n e r a l i z e t h e f i n d i n g t o most of t h e s i t u a t i o n s i n LEC gr o w t h . The l a r g e s t s t r e s s a t a g i v e n c r y s t a l l e n g t h o c c u r s above t h e e n c a p s u l a n t s u r f a c e . I f t h i s 222 F i g u r e 9.40 MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l l e n g t h of 52 mm c o m p a r a b l e t o t h e b o r o n o x i d e t h i c k n e s s of 50 mm. R a d i u s 40 mm. A v e r a g e g r a d i e n t 50°C/cm. 223 224 F i g u r e 9.41 MRSS-CRSS ( Y i e l d ) (MPa) f o r t h r e e a v e r a g e g r a d i e n t s , (a) 43°C/cm, (b) 14°C/cm, (c) 7°C/cm. R a d i u s 40 mm, l e n g t h 80 mm, b o r o n o x i d e t h i c k n e s s 50 mm. 225 p o s i t i o n i s h a l f - w a y a l o n g the c r y s t a l l e n g t h , t h e s t r e s s i s t h e l a r g e s t i n t h e whole growth p e r i o d . The s t r e s s d i s t r i b u t i o n s f o r t h e t h r e e g r a d i e n t s c o n s i d e r e d a r e shown i n F i g u r e 9 . 4 1 ( a - c ) . As i n t h e c a s e of t h e 40 mm e n c a p s u l a n t , t h e g r a d i e n t s h o u l d be f u r t h e r r e d u c e d to o b t a i n s i m i l a r r e l a t i v e a r e a s w i t h z e r o s t r e s s when c o m p a r i n g w i t h t h e 27.5 mm c r y s t a l s d i s t r i b u t i o n s . The e v a l u a t i o n of the r e s u l t s p r e s e n t e d i n t h i s s e c t i o n a r e summarized i n T a b l e 9.4. In t h i s t a b l e t h e l a r g e s t v a l u e o f MRSS- CRSS ( Y i e l d ) i s g i v e n as a f u n c t i o n of b o r o n o x i d e t h i c k n e s s , g r a d i e n t o f the t h e r m a l p r o f i l e and c r y s t a l r a d i u s . The s t r e s s v a l u e s c o r r e s p o n d t o c r y s t a l s w i t h an a s p e c t r a t i o o f 2. F o r e a c h c o n d i t i o n the r e l a t i v e s t r e s s i n c r e m e n t due t o an i n c r e a s e i n r a d i u s i s a l s o r e p o r t e d . I t i s c l e a r from t h e t a b u l a t e d v a l u e s t h a t t h e s t r e s s d e c r e a s e s as more f a v o u r a b l e growth c o n d i t i o n s a p p l y , i . e . t h i c k e r e n c a p s u l a n t and lo w e r g r a d i e n t . I t i s a l s o o b s e r v e d t h a t the r e l a t i v e s t r e s s change due t o an i n c r e a s e i n r a d i u s i n c r e a s e s as t h e s t r e s s l e v e l d e c r e a s e s ( o r more f a v o u r a b l e c o n d i t i o n s a r e p r e s e n t ) . In the c a s e of the t h i c k e s t e n c a p s u l a n t and t h e l o w e s t g r a d i e n t used i n t h e c a l c u l a t i o n s , t h e s t r e s s i n c r e a s e s 143.7 % when i n c r e a s i n g t h e r a d i u s from 27.5 mm to 40 mm. T h i s shows t h a t t h e e f f e c t on s t r e s s e s w i t h i n c r e a s i n g t h e r a d i u s i s not l i n e a r . A c o m p a r i s o n of the s t r e s s c o n t o u r s f o r a l l c o n d i t i o n s l i s t e d i n the t a b l e shows t h a t s i m i l a r s t r e s s d i s t r i b u t i o n s a r e T a b l e 9.4 E f f e c t of Thermal C o n d i t i o n s on S t r e s s e s B o r o n O x i d e AMRSS - CRSS ( Y i e l d ) MPa T h i c k n e s s | G | 1 % [mm] R = 2.75 cm | R = 4.0 cm change G 7.02 9.65 37.46 G/2 3.76 21.0 G/4 1.78 2.76 55.0 G/6 1.03 G/8 0.65 1.02 56.0 40 . 0 G G/2 G/3 G/5 4 . 39 1 . 97 1 . 22 6 . 77 3.21 2.01 1 . 02 54 . 2 62 . 9 64 . 7 50 . 0 G G/2 G/3 G/5 3 . 26 1.14 0 . 64 5 . 61 1 . 56 0 . 47 72.08 143.7 227 o b t a i n e d f o r s i m i l a r v a l u e s of s t r e s s . F o r i n s t a n c e , s i m i l a r a r e a s w i t h z e r o s t r e s s a r e g i v e n i n t h e c r y s t a l s w i t h t h e f o l l o w i n g c o n d i t i o n s 1) B = 21 mm, G/6, r = 2.75 mm ; 2) B = 21 mm, G/8, r = 40 mm ; 3) B = 40 mm, G/5, r = 40 mm. 9.6 Gas P r e s s u r e and C o m p o s i t i o n C o n d i t i o n s : AP, 2 atm ; R, 27.5 mm ; CL, 55 mm ; B, 21 mm ; T e m p e r a t u r e p r o f i l e s as i n F i g u r e 9.33 ; AVG : (a) 66°C/cm, (b) 33°C/cm, (c) 17°C/cm, (d) ll°C/cm. The e f f e c t of gas p r e s s u r e i s c o n s i d e r e d i n the model t h r o u g h t h e e f f e c t of p r e s s u r e on t h e c o n v e c t i v e p a r t of t h e he a t t r a n s f e r c o e f f i c i e n t between c r y s t a l and gas. In the e q u a t i o n s used f o r the n u m e r i c a l e v a l u a t i o n , t h e c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t changes w i t h t h e s q u a r e r o o t o f p r e s s u r e . The r e s u l t s f o r the MRSS-CRSS ( Y i e l d ) a r e shown i n F i g u r e s 9.42(a-d) f o r f o u r g r a d i e n t s . The e f f e c t of p r e s s u r e on the f o u r g r a d i e n t s can be a n a l y s e d by c o m p a r i n g t h e s t r e s s v a l u e s w i t h t h e r e s u l t s o b t a i n e d f o r a p r e s s u r e o f 30 a t m o s p h e r e s . F o r t h i s F i g u r e 9.42(b-c) s h o u l d be compared w i t h F i g u r e s 9 . 3 4 ( a - c ) . I t i s o b s e r v e d t h a t the s t r e s s c o n f i g u r a t i o n f o r the f o u r g r a d i e n t s a t 2 a t m o s p h e r e s i s s i m i l a r t o t h e s t r e s s c o n f i g u r a t i o n f o r t h e same  229 c d .42 MRSS-CRSS ( Y i e l d ) (MPa) f o r f o u r t e m p e r a t u r e p r o f i l e s w i t h a v e r a g e g r a d i e n t s . (a) 33°c/cm, (b) 17°C/cm, (c) 11 C/cm and (d) 8 C/cm. R a d i u s 27.5 mm, l e n g t h 55 mm. E n c a p s u l a n t t h i c k n e s s 21 mm. Argon p r e s s u r e 2 atm . 230 imposed t e m p e r a t u r e p r o f i l e s a t a p r e s s u r e o f 30 a t m o s p h e r e s . At t h e l a r g e s t g r a d i e n t c o n s i d e r e d , t h e s t r e s s d e c r e a s e s o n l y 3 % from 30 to 2 a t m o s p h e r e s . F o r t h e low e r g r a d i e n t s t h e d e c r e a s e i s i n t h e o r d e r o f 20 %. F o r t h e l o w e s t g r a d i e n t c o n s i d e r e d t h e a r e a w i t h z e r o MRSS-CRSS ( Y i e l d ) i n F i g u r e 9.42(d) i n c r e a s e d about 25 % f o l l o w i n g t h e d e c r e a s e i n t h e l a r g e s t s t r e s s v a l u e . The r e s u l t s p r e s e n t e d h e r e show t h a t the e f f e c t o f p r e s s u r e i s n ot as i m p o r t a n t as might be e x p e c t e d . The weak e f f e c t of p r e s s u r e o b s e r v e d i n the p r e s e n t c a l c u l a t i o n s when a s t r o n g e r e f f e c t i s e x p e c t e d , can be a c c o u n t e d f o r i n t h e f o l l o w i n g way. In the model the e f f e c t of p r e s s u r e i s i n c l u d e d as a f f e c t i n g t h e c o n v e c t i v e component of t h e h e a t t r a n s f e r c o e f f i c i e n t as t h e s q u a r e r o o t of p r e s s u r e . R e d u c i n g the p r e s s u r e from 30 t o 2 a t m o s p h e r e s r e d u c e s t h e v a l u e of c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t by 74 % . At low p r e s s u r e s the c o n v e c t i v e p a r t i s l e s s t h a n 10 % o f t h e t o t a l v a l u e of t h e h e a t t r a n s f e r c o e f f i c i e n t . In t h i s c a s e the maximum v a r i a t i o n i n t h e t o t a l c o e f f i c i e n t a s s o c i a t e d w i t h t h e r e d u c t i o n i n p r e s s u r e w i l l be l e s s t h a n 30 * I t was shown i n S e c t i o n 9.1 above, t h a t a 30 % change i n he a t t r a n s f e r c o e f f i c i e n t does not s u b s t a n t i a l l y a f f e c t t h e s t r e s s l e v e l i n t h e c r y s t a l . T h i s e x p l a i n s why t h e e f f e c t of p r e s s u r e i n t h e p r e s e n t c a l c u l a t i o n s i s not s i g n i f i c a n t . A s t r o n g e r e f f e c t of p r e s s u r e i s e x p e c t e d b e c a u s e p r e s s u r e may m a r k e d l y a f f e c t t h e t e m p e r a t u r e p r o f i l e of t h e media s u r r o u n d i n g the c r y s t a l . In t h e p r e s e n t c a l c u l a t i o n s , i t was assumed t h a t c h a n g i n g t h e p r e s s u r e does not a l t e r t h e t e m p e r a t u r e 231 o f t h e e n v i r o n m e n t . T h i s may not be the a c t u a l s i t u a t i o n . When the p r e s s u r e c h a n g e s , not o n l y t h e h e a t t r a n s f e r c o e f f i c i e n t a t the c r y s t a l - g a s i n t e r f a c e changes but a l s o t h e e f f i c i e n c y i n h e a t t r a n s f e r i n t h e gas i s a l t e r e d . T h i s h i g h e r e f f i c i e n c y w i l l be m a n i f e s t e d by a l a r g e r h e a t t r a n s f e r c o e f f i c i e n t a t the b o r o n o x i d e - g a s i n t e r f a c e . The net e f f e c t w i l l be t o d e c r e a s e t h e t e m p e r a t u r e o f t h e b o r o n o x i d e s u r f a c e i n c r e a s i n g the g r a d i e n t a c r o s s t h e e n c a p s u l a n t and p r o b a b l y i n t h e g a s . The change i n heat t r a n s f e r c o e f f i c i e n t a t t h e b o r o n o x i d e - gas s u r f a c e can be q u a l i t a t i v e l y d e s c r i b e d , a s s u m i n g a f r e e c o n v e c t i o n mechanism a p p l i e s a t t h i s s u r f a c e . In t h i s c a s e t h e h e a t t r a n s f e r c o e f f i c i e n t a s s o c i a t e d w i t h t h i s p r o c e s s can be w r i t t e n as I / O h = Nu k/1 = 0.14 (Gr P r ) ' (k/1) (9.1) where Nu, Gr and Pr a r e t h e N u s s e l t , G r a s h o f and P r a n d l numbers r e s p e c t i v e l y . "k" i s t h e t h e r m a l c o n d u c t i v i t y . "1" i s t h e c h a r a c t e r i s t i c l e n g t h of the h o r i z o n t a l h e a t e d s u r f a c e from w h i c h h e a t i s e x t r a c t e d . I f i t i s assumed t h a t t h e gas behaves l i k e an i d e a l g as, t h e p r e s s u r e dependence i s i n t r o d u c e d i n the G r a s h o f number t h r o u g h t h e d e n s i t y . In t h i s c a s e t h e h e a t t r a n s f e r c o e f f i c i e n t w i l l depend on p r e s s u r e w i t h an e x p o n e n t 2/3. F o r t h i s p r e s s u r e dependence th e h e a t t r a n s f e r c o e f f i c i e n t i s r e d u c e d by 83 % when d e p r e s s u r i z i n g the chamber from 30 t o 2 a t m o s p h e r e s . At t h e e n c a p s u l a n t - g a s s u r f a c e t h i s change i s more i m p o r t a n t t h a n s i m i l a r changes o f the c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t a t 232 t h e c r y s t a l - g a s s u r f a c e s i n c e i n t h e f i r s t c a s e c o n v e c t i o n i s t h e main h e a t t r a n s f e r mechanism. T h i s i s b e c a u s e b o r o n o x i d e i s t r a n s p a r e n t t o r a d i a t i o n . C o n s e q u e n t l y t h e t h e r m a l g r a d i e n t a c r o s s t h e e n c a p s u l a n t w i l l be more d i r e c t l y a f f e c t e d by p r e s s u r e c h a n g e s . On t h e o t h e r hand, i n c r e a s i n g the p r e s s u r e i n c r e a s e s the h e a t t r a n s f e r c o e f f i c i e n t at t h e c r y s t a l - g a s s u r f a c e . In t h i s c a s e h e a t e x t r a c t i o n from t h e c r y s t a l i s i m p r o v e d r e d u c i n g t h e r m a l g r a d i e n t s i n the gas. The n e t e f f e c t o f i n c r e a s i n g t h e c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t at t h e c r y s t a l s u r f a c e and i n c r e a s i n g t h e g r a d i e n t s a c r o s s t h e e n c a p s u l a n t c o u l d r e s u l t i n a s t r o n g e r e f f e c t i n the s t r e s s d i s t r i b u t i o n i n t h e c r y s t a l t h a n any c h a n ges i n t h e c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t . 14 3-144 The e x p e r i m e n t a l r e s u l t s o f Emori e t a l . a r e i n agreement w i t h the e f f e c t of p r e s s u r e on t h e t h e r m a l g r a d i e n t a c r o s s t h e e n c a p s u l a n t d i s c u s s e d above. F o r a r g o n gas, r e d u c i n g 2 t h e p r e s s u r e from 20 to 5 kg/cm r e d u c e s t h e t h e r m a l g r a d i e n t i n t h e e n c a p s u l a n t from a p p r o x i m a t e l y 160°C/cm t o 140°C/cm ; the t h i c k n e s s o f e n c a p s u l a n t i s not g i v e n . F o r neon and h e l i u m g a s e s l a r g e r g r a d i e n t s were measured, w i t h neon h a v i n g t h e l a r g e s t g r a d i e n t . C h a n g i n g gas p r e s s u r e i n neon has a s t r o n g e r e f f e c t on g r a d i e n t s t h a n i n h e l i u m . They have e x p l a i n e d t h e r e s u l t s by c a l c u l a t i n g t h e h e a t t r a n s f e r c o e f f i c i e n t s f o r the d i f f e r e n t g a s e s and p r e s s u r e s . F o r t h e c a l c u l a t i o n h e a t t r a n s f e r from a v e r t i c a l w a l l was assumed. In t h i s c a s e t h e h e a t t r a n s f e r c o e f f i c i e n t e q u a t i o n i s w r i t t e n as h = 0.53 (k/1) (Gr P r ) 1 / 4 (9.2) 2 33 T h i s e q u a t i o n i n v o l v e s the same q u a n t i t i e s as i n E q u a t i o n ( 9 . 1 ) . The d i f f e r e n c e s a r e t h e v a l u e s o f t h e n u m e r i c a l c o e f f i c i e n t s , t h e e x p o n e n t s a f f e c t i n g t h e q u a n t i t i e s and t h e meaning o f t h e c h a r a c t e r i s t i c l e n g t h "1". In t h i s c a s e "1" i s t h e l e n g t h of t h e c r y s t a l w hich a c t s as a v e r t i c a l w a l l . The c a l c u l a t i o n s show a v e r y good c o r r e l a t i o n between the g r a d i e n t i n t h e b o r o n o x i d e f o r t h e d i f f e r e n t g a s e s and p r e s s u r e s w i t h t h e h e a t t r a n s f e r c o e f f i c i e n t v a l u e s . T h i s c o r r e l a t i o n i s r e p r o d u c e d i n F i g u r e 9.43, c u r v e A. The c o r r e l a t i o n , however, i s b a s i c a l l y i n c o n s i s t e n t s i n c e the measurements were made w i t h o u t a c r y s t a l b e i n g p r e s e n t . M o r e o v e r , i n t r o d u c i n g t h e c r y s t a l and u s i n g a l a r g e r h e a t t r a n s f e r c o e f f i c i e n t w i l l t e n d t o r e d u c e t h e t e m p e r a t u r e g r a d i e n t a c r o s s t h e e n c a p s u l a n t as e x p l a i n e d above. F o r t u i t o u s l y , t h e c o r r e l a t i o n i n F i g u r e 9.43 ( c u r v e A) g i v e s t h e c o r r e c t t e n d e n c y but t h r o u g h a d i f f e r e n t mechanism. The gas p r e s s u r e a f f e c t s t h e t e m p e r a t u r e g r a d i e n t a c r o s s t h e e n c a p s u l a n t t h r o u g h changes i n the h e a t t r a n s f e r c o e f f i c i e n t a t t h e e n c a p s u l a n t - g a s s u r f a c e . A c o r r e l a t i o n between h e a t t r a n s f e r c o e f f i c i e n t v a l u e s f o r h e a t t r a n s f e r from an h o r i z o n t a l w a l l 14 3 14 4 ( E q u a t i o n ( 9 . 1 ) ) and measured g r a d i e n t s from E m o r i et a l . ' g i v e s the r e s u l t s shown i n F i g u r e 9.43, c u r v e B. In t h i s c a s e a d e v i a t i o n from l i n e a r i t y i s o b s e r v e d . The c o r r e l a t i o n , however, i s c o n s i s t e n t w i t h the p h y s i c a l p r o c e s s a n a l y s e d . From the model r e s u l t s and t h e above d i s c u s s i o n i t i s p o s s i b l e to c o n c l u d e t h a t the gas p r e s s u r e and gas c o m p o s i t i o n 234 V e r t i c a l Heat T r a n s f e r Coe f f i c ien t (W/cm K) £ u O c cu •o o v . o 5 i d 3 10 i d 3 2 00 1 1 1 1 1 1 1 1 1 / s , / ^^^^ 1 1 1 _ A V ^x - <r sY 1 1 1 1 1 1 1 1 1 1 1 0 I 2 Hor izonta l Heot T r a n s f e r C o e f f i c i e n t ( A r b i t r a r y U n i t s ) G a s Ar N 2 He P re s su re k g / c m ^ 20 , 5 2 0 , 5 20 , 5 V e r t i c o l o & O Hor i zon ta l F i g u r e 9.43 C o r r e l a t i o n between t h e measured t e m p e r a t u r e g r a d - i e n t s a c r o s s the e n c a p s u l a n t and t h e h e a t t r a n s f e r c o e f f i c i e n t v a l u e s as a f u n c t i o n of gas p r e s s u r e and gas n a t u r e . Curve A from R e f s . 143-144 u s i n g c o n v e c t i o n from a v e r t i c a l w a l l ; and c u r v e B u s i n g c o n v e c t i o n from an h o r i z o n t a l s u r f a c e . 2 3 5 may a f f e c t the s t r e s s d i s t r i b u t i o n i n the c r y s t a l . A change i n t h e h e a t t r a n s f e r c o e f f i c i e n t between the c r y s t a l and t h e gas i s not enough to e x p l a i n the l a r g e e f f e c t on d i s l o c a t i o n d e n s i t y c h a n g e s a s s o c i a t e d w i t h gas p r e s s u r e and c o m p o s i t i o n . A f u l l d e s c r i p t i o n o f t h e e f f e c t of t h e s e v a r i a b l e s r e q u i r e s t h e i n c l u s i o n of t h e h e a t t r a n s f e r mechanisms a t t h e e n c a p s u l a n t - g a s i n t e r f a c e . A d e c r e a s e i n gas p r e s s u r e d e c r e a s e s the h e a t f l u x t o t h e gas a t t h i s s u r f a c e w h i c h r e s u l t s i n s m a l l e r t h e r m a l g r a d i e n t s a c r o s s th e e n c a p s u l a n t . Such an e n v i r o n m e n t w i l l g i v e l o w e r s t r e s s e s i n t h e c r y s t a l . S i n c e th e t e m p e r a t u r e p r o f i l e i n t h e e n v i r o n m e n t i s i n p u t i n the model and r e l i a b l e d a t a on t h e e f f e c t of gas p r e s s u r e and c o m p o s i t o n i s not a v a i l a b l e , p r e c i s e c a l c u l a t i o n s on t h i s m a t t e r were not done. 9.7 C u r v a t u r e of the S o l i d - L i q u i d I n t e r f a c e The e f f e c t o f a n o n - p l a n a r s o l i d - l i q u i d i n t e r f a c e on t h e s t r e s s e s i s s t u d i e d c o n s i d e r i n g convex and c o n c a v e i n t e r f a c e s h a p e s w i t h r e s p e c t to t h e m e l t . In b o t h c a s e s t h e i n t e r f a c e i s assumed to be a p a r a b o l o i d o f r e v o l u t i o n w hich i s r e p r e s e n t e d by t h e f o l l o w i n g a n a l y t i c a l f u n c t i o n s £ = ± 0 . 3 p 2 p and 5 a r e t h e n o n d i m e n s i o n a l r a d i a l and a x i a l p o s i t i o n r e s p e c t i v e l y w i t h the o r i g i n s i t u a t e d at t h e a x i s o f t h e c r y s t a l ; t h e + and - s i g n s c o r r e s p o n d t o t h e convex and c o n c a v e i n t e r f a c e s r e s p e c t i v e l y . The v a l u e o f t h e c o n s t a n t 0 . 3 g i v e s th e 236 r e l a t i v e d i s p l a c e m e n t between t h e c e n t e r and edge o f t h e i n t e r f a c e . The mesh used i n t h i s c a s e i s shown i n F i g u r e 9.44(a) f o r a convex and (b) f o r a c o n c a v e i n t e r f a c e . They a r e g e n e r a t e d m a n u a l l y . The e f f e c t o f a number o f growth c o n d i t i o n s combined w i t h the i n t e r f a c e shape i s c o n s i d e r e d . The most i m p o r t a n t o b s e r v a t i o n s f o l l o w . 9.7.1 Convex i n t e r f a c e The e f f e c t o f a convex i n t e r f a c e combined w i t h t h e e f f e c t o f c r y s t a l l e n g t h , t h e r m a l c o n d i t i o n s and c r y s t a l r a d i u s i s c o n s i d e r e d . 9.7.1.1 C r y s t a l L e n g t h C o n d i t i o n s : R, 27.5 mm ; CA, 3 0 ° ; AP, 30 atm. ; B , 21 mm ; T e m p e r a t u r e p r o f i l e as i n F i g u r e 8.19, c u r v e 4, CL : (a) 27.5 mm = R, (b) 55.0 mm = 2 R, (c) 82.5 mm = 3 R . The t e m p e r a t u r e and Von M i s e s s t r e s s f i e l d s f o r a c r y s t a l l e n g t h o f 55 mm a r e shown i n F i g u r e 9.45. In t h e r i g h t p a r t of the f i g u r e t h e t e m p e r a t u r e f i e l d i s d i f f e r e n t t h a n t h a t f o r a c r y s t a l w i t h a f l a t i n t e r f a c e . In t h e lower h a l f of t h e c r y s t a l t h e i s o t h e r m s p r e s e n t a l a r g e c u r v a t u r e f o l l o w i n g t h e i n t e r f a c e shape and g i v i n g n e g a t i v e r a d i a l g r a d i e n t s . In t h i s p a r t o f t h e c r y s t a l t h e a x i a l g r a d i e n t s a r e l a r g e r a t the c r y s t a l edge t h a n 237 F i g u r e 9.44 Element and nodal c o n f i g u r a t i o n at the i n t e r f a c e f o r c r y s t a l s with curved i n t e r f a c e , (a) Convex i n t e r f a c e , (b) Concave i n t e r f a c e . 238 a t t h e a x i s . The a x i a l g r a d i e n t s a t t h e a x i s do not change w i t h d i s t a n c e from t h e i n t e r f a c e . The Von M i s e s s t r e s s f i e l d w h i c h i s shown i n the r i g h t p a r t o f F i g u r e 9.45 i s d i f f e r e n t t h a n t h e d i s t r i b u t i o n o b t a i n e d w i t h a p l a n a r i n t e r f a c e . The l a r g e s t VMS i s o b s e r v e d t o be i n two r e g i o n s i n t h e c r y s t a l , below t h e c r y s t a l s h o u l d e r and a t the edge o f t h e s o l i d - l i q u i d i n t e r f a c e . The o r i g i n and n a t u r e of t h e f i r s t s t r e s s c o n c e n t r a t i o n was d i s c u s s e d p r e v i o u s l y f o r t h e c a s e of a p l a n a r i n t e r f a c e . The magni t u d e o f t h e VMS i n t h i s r e g i o n i s about 14 * l e s s t h a n f o r a p l a n a r i n t e r f a c e . The s t r e s s c o n c e n t r a t i o n a t t h e i n t e r f a c e edge, w h i c h i s u n i q u e f o r a c u r v e d i n t e r f a c e , shows s t r e s s v a l u e s w h i c h a r e 30 t o 40 * l a r g e r t h a n the s t r e s s below t h e s h o u l d e r . On t h e o t h e r hand t h e r e g i o n w i t h low s t r e s s e s , w h i c h f o r a p l a n a r i n t e r f a c e i s a t t h e i n t e r f a c e edge, has moved t o t h e c e n t e r and low e r p a r t o f t h e c r y s t a l n e a r t h e a x i s . The l o w e s t s t r e s s e s a r e t h r e e t i m e s s m a l l e r t h a n f o r a p l a n a r i n t e r f a c e . The l a r g e s t s t r e s s at the i n t e r f a c e edge i s a s s o c i a t e d w i t h the l a r g e r a d i a l and a x i a l g r a d i e n t s i n t h a t r e g i o n . The l o w e s t s t r e s s c l o s e t o t h e c r y s t a l a x i s and ab o u t 2 cm from t h e i n t e r f a c e i s a s s o c i a t e d w i t h the n e a r l y c o n s t a n t g r a d i e n t i n t h i s r e g i o n . The MRSS c o n t o u r s f o r t h e t h r e e c r y s t a l l e n g t h s a r e shown i n F i g u r e 9 . 4 6 ( a - c ) . Comparing the r i g h t p a r t o f F i g u r e 9.45 and F i g u r e 9 . 4 6 ( a ) , i t i s o b s e r v e d t h a t t h e VMS and MRSS r e l a t i v e 239 F i g u r e 9.45 T e m p e r a t u r e (10 C) and Von M i s e s s t r e s s (MPa) f i e l d s f o r a c r y s t a l w i t h a convex i n t e r f a c e s hape. R a d i u s 27.5 mm. L e n g t h 27.5 mm. E n c a p s u l a n t t h i c k n e s s 21 mm. T e m p e r a t u r e p r o f i l e as shown i n F i g u r e 8.19, c u r v e 4. 240 F i g u r e 9.46 MRSS (MPa) c o n t o u r s f o r a c r y s t a l w i t h convex i n t e r - f a c e at t h r e e l e n g t h s , (a) 27.5 mm, (b) 55 mm, ( c ) 82.5 mm. The l a r g e s t s t r e s s e s a t t h e i n t e r f a c e edge a r e (a) 7.22 MPa, (b) 9.19 MPa, ( c ) 9.66 MPa. R a d i u s 27.5 mm, Boron o x i d e t h i c k n e s s 21mm. P r o f i l e as i n F i g u r e 8.19, c u r v e 4. 241 d i s t r i b u t i o n s a r e s i m i l a r . The a b s o l u t e l e v e l s f o r the MRSS a r e a p p r o x i m a t e l y h a l f of the c o r r e s p o n d i n g VMS at any p o i n t . The e f f e c t o f c r y s t a l l e n g t h i n the MRSS d i s t r i b u t i o n i s shown i n F i g u r e 9 . 4 6 ( a - c ) . C o m p a r i n g t h e t h r e e d i s t r i b u t i o n s i t i s p o s s i b l e t o d i s t i n g u i s h two r e g i o n s w i t h s p e c i f i c and r e p r o d u c i b l e c h a r a c t e r i s t i c s . One i s t h e r e g i o n c l o s e t o t h e i n t e r f a c e w h i c h e x t e n d s t o a d i s t a n c e o f a p p r o x i m a t e l y a r a d i u s from t h e i n t e r f a c e a t t h e a x i s . The s e c o n d i s t h e c r y s t a l above t h i s r e g i o n . The s t r e s s d i s t r i b u t i o n , i n the low e r p a r t o f t h e c r y s t a l , e x p e r i e n c e s l i t t l e change w i t h c r y s t a l l e n g t h and i s c h a r a c t e r i s t i c o f the convex i n t e r f a c e s hape. The r e g i o n above t h i s has a s t r e s s d i s t r i b u t i o n s i m i l a r t o t h e d i s t r i b u t i o n i n a c r y s t a l w i t h f l a t i n t e r f a c e . From t h i s i t i s c o n c l u d e d t h a t t h e e f f e c t of t h i s convex i n t e r f a c e does not e x t e n d s f u r t h e r t h a n a l e n g t h c o m p a r a b l e to the c r y s t a l r a d i u s . The l a r g e s t s t r e s s v a l u e a t t h e i n t e r f a c e edge i n c r e a s e s w i t h c r y s t a l l e n g t h from (a) t o (b) and s l i g h t l y i n c r e a s e s from (b) t o ( c ) . T h i s l a s t e f f e c t i s a t t r i b u t e d t o a s l i g h t i n c r e a s e i n a x i a l t e m p e r a t u r e g r a d i e n t . The s t r e s s d i s t r i b u t i o n s o f t h e MRSS-CRSS ( Y i e l d ) i s shown i n F i g u r e 9 . 4 7 ( a - c ) f o r the t h r e e c r y s t a l l e n g t h s . In (a) t h e r e a r e two r e g i o n s w i t h z e r o s t r e s s w h i c h a r e a s s o c i a t e d w i t h t h e r e g i o n s w i t h t h e l o w e s t s t r e s s e s from F i g u r e 9 . 4 6 ( a ) . T h i s z e r o s t r e s s r e g i o n a l s o a p p e a r s i n (b) and ( c ) . In ( c ) t h e upper p a r t o f t h e c r y s t a l shows a l a r g e a r e a w i t h z e r o s t r e s s . 242 9.47 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s f o r a c r y s t a l w i t h convex i n t e r f a c e at t h r e e l e n g t h s (a) 27.5 mm, (b) 55 mm, ( c ) 82.5 mm. R a d i u s 27.5 mm, b o r o n o x i d e t h i c k n e s s 21 mm. P r o f i l e as i n F i g u r e 8.19, c u r v e 4. 243 The d i s l o c a t i o n d e n s i t y i n t h e c r y s t a l a t t h e end of growth depends on t h e p o s i t i o n i n the c r y s t a l . The l a r g e MRSS-CRSS ( Y i e l d ) i n F i g u r e 9 . 4 7 (a-c) a t the i n t e r f a c e edge s u g g e s t s t h a t t h e d i s l o c a t i o n s at t h e c r y s t a l s u r f a c e a r e for m e d r i g h t a f t e r s o l i d i f i c a t i o n and not above the e n c a p s u l a n t s u r f a c e as w i t h the p l a n e i n t e r f a c e . The a x i a l MRSS-CRSS ( Y i e l d ) n e a r t h e c r y s t a l s u r f a c e i s l o w e s t c l o s e t o the cone, g r a d u a l l y i n c r e a s i n g w i t h d i s t a n c e from t h e cone. At a d i s t a n c e o f 50 mm from t h e cone t h e MRSS-CRSS ( Y i e l d ) i s d o u b l e t h a t c l o s e t o t h e cone. At the c e n t r e , t h e l a r g e s t s t r e s s e s at a d i s t a n c e o f a r a d i u s from t h e i n t e r f a c e i s h a l f the s t r e s s a t the edge. At t h e s e e d and bottom ends of t h e c r y s t a l the s t r e s s e s a r e low and n e v e r r e a c h t h e maximum s t r e s s e s . At the seed end, the maximum s t r e s s i s a l w a y s l e s s t h a n 3.0 MPa t h a t i s , h a l f t h e v a l u e at t h e edge. S i m i l a r r e s u l t s a r e o b t a i n e d at the t a i l end where t h e maximum s t r e s s a t the c e n t r e o c c u r s v e r y c l o s e t o t h e i n t e r f a c e . Thus c r o s s - s e c t i o n s a t any d i s t a n c e from t h e cone w i l l have a W- sha p e d MRSS-CRSS ( Y i e l d ) d i s t r i b u t i o n a l o n g a d i a m e t e r e x c e p t at t h e t a i l end where the MRSS-CRSS ( Y i e l d ) d i s t r i b u t i o n i s U- sha p e d . The edge o f t h e U i s t h r e e t i m e s h i g h e r t h a n t h e c e n t r e . The symmetry o f t h e MRSS-CRSS ( Y i e l d ) i n (001) p l a n e s depends on t h e p o s i t i o n of the p l a n e w i t h r e s p e c t t o t h e cone. C l o s e t o the cone the MRSS-CRSS ( Y i e l d ) has t h e symmetry shown i n F i g u r e 9.48. T h i s c r o s s - s e c t i o n i s a t a d i s t a n c e o f 11.0 mm from the cone when t h e c r y s t a l i s 27.5 mm l o n g ( F i g u r e 9.47(a) ). The maximum s t r e s s i s r e a c h e d a t t h i s p o i n t . 244 F i g u r e 9.48 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n a (001) p l a n e a t a d i s t a n c e of 11 mm from t h e cone i n t h e c r y s t a l shown i n F i g u r e 9 . 4 7 ( a ) . 245 The symmetry of the MRSS-CRSS ( Y i e l d ) i n s e c t i o n s w h i c h a r e more t h a n 55 mm f a r from t h e ends of t h e c r y s t a l a t t h e end o f gro w t h w i l l be d e t e r m i n e d by t h e sequence o f symmetry changes shown i n F i g u r e 9 . 4 9 ( a - c ) . D e s p i t e the f a c t t h a t t h e s e s e c t i o n s were t a k e n from o n l y one c r y s t a l ( F i g u r e 9 . 4 7 ( c ) ) , i t was o b s e r v e d t h a t a l l t h e s e c t i o n s f a r from the ends p a s s t h r o u g h t h e same symmetry p a t t e r n s and s t r e s s l e v e l s . C l o s e t o t h e i n t e r f a c e ( p a r t (a) ) t h e r e i s f o u r - f o l d symmetry w i t h an U - d i s t r i b u t i o n a l o n g a d i a m e t e r . At the edge e i g h t - f o l d symmetry i s p r e s e n t . In t h i s r e g i o n t h e d i s l o c a t i o n s a r e e x p e c t e d t o have t h e same symmetry. S e c t i o n s from the t a i l end show a symmetry o f t h e d i s l o c a t i o n s as shown i n F i g u r e 9 . 4 9 ( a ) . 9.7.1.2 E f f e c t o f E n c a p s u l a n t T h i c k n e s s and G r a d i e n t C o n d i t i o n s : B, 50 mm ; T e m p e r a t u r e p r o f i l e s as i n F i g u r e 9 . 3 5 ( b ) , The r e s u l t s f o r the s t r e s s d i s t r i b u t i o n s f o r two g r a d i e n t s a r e shown i n F i g u r e 9.50(a-b) f o r a 50°C/cm g r a d i e n t and F i g u r e 9.51(a-b) f o r a 12.5°C/cm g r a d i e n t . P a r t (a) i n each f i g u r e c o r r e s p o n d s to t h e MRSS d i s t r i b u t i o n and p a r t (b) t o the MRSS- CRSS ( Y i e l d ) d i s t r i b u t i o n . The MRSS d i s t r i b u t i o n s f o r t h e two g r a d i e n t s show a c o n c e n t r a t i o n o f s t r e s s a t the i n t e r f a c e edge and below the s h o u l d e r . The s t r e s s a t t h e i n t e r f a c e edge i s about 80 % l a r g e r t h a n below t h e s h o u l d e r . The l o w e s t s t r e s s l e v e l o c c u r s a t the 246 a 247 248 F i g u r e 9.49 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n (001) p l a n e s a t t h r e e d i s t a n c e s form t h e cone. (a) 79.75 mm, (b) 66.0 mm, ( c ) 52.25 mm. S e c t i o n c o r r e s p o n d s to the c r y s t a l shown i n F i g u r e 9 . 4 7 ( c ) . 249 F i g u r e 9.50 (a) MRSS (MP) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l w i t h convex i n t e r f a c e . Boron o x i d e t h i c k n e s s 50 mm. R a d i u s 27.5 mm. L e n g t h 55 mm. A v e r a g e g r a d i e n t 50°c/cm. 250 F i g u r e 9.51 (a) MRSS (MP) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l w i t h convex i n t e r f a c e . Boron o x i d e t h i c k n e s s 50 mm. R a d i u s 27.5 mm. ' L e n g t h 55 mm. A v e r a g e g r a d i e n t 17°C/cm. 251 c e n t r e o f t h e c r y s t a l . The l a r g e s t s t r e s s v a l u e a t t h e i n t e r f a c e edge d e c r e a s e s p r o p o r t i o n a l l y w i t h a r e d u c t i o n i n g r a d i e n t . The MRSS-CRSS ( Y i e l d ) i n p a r t (b) of the f i g u r e s f o r t h e two g r a d i e n t s shows z e r o s t r e s s r e g i o n s . The a r e a of t h i s r e g i o n s u b s t a n t i a l l y i n c r e a s e s w i t h t h e g r a d i e n t . F o r b o t h c o n d i t i o n s l a r g e a r e a s c l o s e t o t h e i n t e r f a c e edge show s t r e s s e s w h i c h a r e l a r g e r t h a n the CRSS ( Y i e l d ) . 9.7.1.3 E f f e c t of c r y s t a l R a d i u s C o n d i t i o n s : R, 40 mm ; CL, 80 mm ; B, 21 mm ; T e m p e r a t u r e p r o f i l e a i n F i g u r e 8.19, c u r v e 4. The MRSS and the MRSS-CRSS ( Y i e l d ) f o r t h i s c r y s t a l a r e shown i n F i g u r e 9.52(a) and (b) r e s p e c t i v e l y . The s t r e s s d i s t r i b u t i o n s a r e s i m i l a r to t h o s e o b t a i n e d w i t h the 27.5 mm c r y s t a l w i t h two common c h a r a c t e r i s t i c s . L a r g e s t r e s s l e v e l s e x i s t a t the i n t e r f a c e edge and above t h e e n c a p s u l a n t s u r f a c e . The l a r g e s t s t r e s s o c c u r s a t t h e i n t e r f a c e edge. I t i s i m p o r t a n t t o n o t e t h a t w h i l e the s t r e s s l e v e l above the e n c a p s u l a n t s u r f a c e i n c r e a s e s as t h e r a d i u s i n c r e a s e s (45 % ) , t h e s t r e s s l e v e l a t t h e i n t e r f a c e edge o n l y i n c r e a s e s a bout 10 %. On t h e o t h e r hand the symmetry of s t r e s s e s i n (001) p l a n e s show t h e same f e a t u r e s as d e s c r i b e d f o r t h e 27.5 mm c r y s t a l . 252 253 9.7.2 C oncave I n t e r f a c e C o n d i t i o n s B, 21 mm T e m p e r a t u r e p r o f i l e as i n F i g u r e 8.19, c u r v e 4, a) R, 27.5 mm CL, 55 mm b) R, 40 mm CL, 80 mm. The r e s u l t s f o r the MRSS-CRSS ( Y i e l d ) f o r t h e two c r y s t a l r a d i i a r e shown i n F i g u r e 9.53(a) and ( b ) . P a r t (a) f o r t h e 27.5 mm r a d i u s and p a r t (b) f o r t h e 40 mm r a d i u s . As i n t h e c a s e of t h e convex i n t e r f a c e , t h e s t r e s s d i s t r i b u t i o n may be d i v i d e d i n two r e g i o n s . One c l o s e t o t h e i n t e r f a c e and e x t e n d i n g a d i s t a n c e from the i n t e r f a c e c o m p a r a b l e to t h e r a d i u s and a n o t h e r formed by t h e r e g i o n above t h i s s e c t i o n . The low r e g i o n has s t r e s s d i s t r i b u t i o n s w h i c h a r e s i m i l a r f o r b o t h c r y s t a l r a d i i and a r e c h a r a c t e r i s t i c o f t h e c o n c a v e i n t e r f a c e shape. V e r y l a r g e s t r e s s e s a r e p r e s e n t a t t h e i n t e r f a c e w i t h l a r g e s t s t r e s s a t t h e c e n t r e o f t h e c r y s t a l . In t h e s e c o n d r e g i o n , w e l l above the i n t e r f a c e , the s t r e s s d i s t r i b u t i o n i s s i m i l a r to t h a t o b t a i n e d w i t h a p l a n a r i n t e r f a c e . The l a r g e s t s t r e s s e s o c c u r a t t h e i n t e r f a c e and t h e r e f o r e t h e d i s l o c a t i o n d e n s i t y i s c o n t r o l l e d by t h e s t r e s s d i s t r i b u t i o n a t t h e c o n c a v e i n t e r f a c e . F o r t h i s i n t e r f a c e shape, the s t r e s s d i s t r i b u t i o n i n t h e r a d i a l d i r e c t i o n f o l l o w s t h e W shape w i t h the c e n t r e a b o u t 15 * h i g h e r t h a n the e d g e s . The e f f e c t o f i n c r e a s i n g the r a d i u s f o r a c o n c a v e i n t e r f a c e i s d i f f e r e n t t h a n t h a t o b t a i n e d f o r a convex i n t e r f a c e . F o r 254 F i g u r e 9.53 (a) MRSS (MP) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l w i t h c o n c a v e i n t e r f a c e , (a) R a d i u s 27.5 mm, (b) 40 mm. Boron o x i d e t h i c k n e s s 21 mm. 255 the c o n c a v e i n t e r f a c e i n c r e a s i n g t h e r a d i u s a b o u t 45 % i n c r e a s e s t h e l a r g e s t s t r e s s v a l u e from 14.26 MPa t o 18.86 MPa (30 * ) . The symmetry o f the d i s l o c a t i o n s formed c l o s e to t h e i n t e r f a c e i s shown i n F i g u r e 9.54. T h i s (001) p l a n e c o r r e s p o n d s to a s e c t i o n at 2.75 mm from t h e i n t e r f a c e f o r t h e 27.5 mm c r y s t a l . S i m i l a r symmetry i s o b s e r v e d f o r t h e 40 mm c r y s t a l . In t h e c e n t r a l r e g i o n e i g h t - f o l d symmetry i s o b s e r v e d . F o u r - f o l d symmetry i s a l s o o b s e r v e d i n a r i n g c l o s e t o t h e edge. In t h i s c a s e t y p i c a l e i g h t - f o l d a t t h e edge i s not o b s e r v e d . 256 F i g u r e 9.54 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n a (001) p l a n e a t a d i s t a n c e o f 2.75 mm from t h e i n t e r f a c e i n t h e c r y s t a l shown i n F i g u r e 9 . 5 3 ( a ) . 257 CHAPTER 10 RESULTS FOR COOLING AFTER GROWTH U s i n g t h e combined a n a l y t i c a l and n u m e r i c a l scheme d e s c r i b e d i n C h a p t e r 7 and e v a l u a t e d i n C h a p t e r 8 t h e t e m p e r a t u r e and s t r e s s f i e l d s d u r i n g c o o l i n g a f t e r g r o w t h a r e c a l c u l a t e d . R e s u l t s a r e o b t a i n e d f o r a c r y s t a l w i t h a r a d i u s 27.5 mm and a l e n g t h o f 110.0 mm. T h i s l e n g t h c o r r e s p o n d s to t h e l a r g e s t l e n g t h f o r w h i c h s t r e s s and t e m p e r a t u r e c a l c u l a t i o n s a r e made d u r i n g g r o w t h . The a x i a l t e m p e r a t u r e at t h e end o f growth i s t h e i n i t i a l t e m p e r a t u r e f i e l d f o r t h e c o o l i n g model w i t h t h e r a d i a l g r a d i e n t n e g l e c t e d . In t h e c a l c u l a t i o n o f t h e t h e r m a l and s t r e s s f i e l d s i n t h e c r y s t a l a f t e r s o l i d i f i c a t i o n , the i n i t i a l t e m p e r a t u r e p r o f i l e i n t h e c r y s t a l , and t h e t h e r m a l c h a r a c t e r i s t i c s of t h e media s u r r o u n d i n g t h e c r y s t a l a r e of major s i g n i f i c a n c e . In t h e c a l c u l a t i o n s , t h r e e t e m p e r a t u r e g r a d i e n t s i n t h e r e g i o n o f l e n g t h R above th e i n t e r f a c e a r e c o n s i d e r e d , 70, 35 and 17.5°C/cm d e s i g n a t e d as GC , GC/2 and GC/4. These g r a d i e n t s a r e a b o u t h a l f t h e g r a d i e n t s i n t h e g r o w i n g chamber d u r i n g s o l i d i f i c a t i o n . The c r y s t a l as i t c o o l s , i s s u r r o u n d e d by e i t h e r a r g o n a t 30 a t m o s p h e r e s or b o r o n o x i d e . The a r g o n t e m p e r a t u r e s c o n s i d e r e d a r e 800 and 1000°C and t h e b o r o n o x i d e t e m p e r a t u r e i s t a k e n as 1000°C . 258 10.1 Ambient T e m p e r a t u r e 1000°C 10.1.1. Ar g o n 10.1.1.1. I n i t i a l G r a d i e n t GC = 70°C/cm The t e m p e r a t u r e d i s t r i b u t i o n o v e r t h e r i g h t h a l f of a v e r t i c a l s e c t i o n o f a c r y s t a l a f t e r 5 s e c o n d s from s o l i d i f i c a t i o n i s shown i n F i g u r e 1 0 . 1 ( a ) . The r e s u l t s show t h a t i n a r e g i o n of a b o u t 1 cm a d j a c e n t to the i n t e r f a c e , the t e m p e r a t u r e has d r o p p e d r a p i d l y . The a x i a l g r a d i e n t i n t h i s r e g i o n changes from p o s i t i v e t o n e g a t i v e . The r a d i a l g r a d i e n t s a r e h i g h , and t h e t e m p e r a t u r e d r o p at t h e edge i s l a r g e , a b o u t 80°C. The r a d i a l g r a d i e n t i s n e g a t i v e between the i n t e r f a c e and the 1000°C i s o t h e r m , and t h e n becomes p o s i t i v e a t lower t e m p e r a t u r e . With a c h a n g i n g ambient t e m p e r a t u r e , w h i c h i s n o r m a l l y t h e c a s e , ambient t e m p e r a t u r e s w i l l be lower t h a n the c r y s t a l s u r f a c e t e m p e r a t u r e s r e s u l t i n g i n n e g a t i v e r a d i a l g r a d i e n t s t h r o u g h o u t t h e c r y s t a l . The a s s u m p t i o n o f c o n s t a n t ambient t e m p e r a t u r e i n t h e model w i l l l i k e l y g i v e c a l c u l a t e d s t r e s s l e v e l s w h i c h a r e t o o low. T h i s e r r o r w i l l become l e s s s i g n i f i c a n t as t h e ambient t e m p e r a t u r e g r a d i e n t s become s m a l 1 e r . The t e m p e r a t u r e d i s t r i b u t i o n 10 s e c o n d s a f t e r s o l i d i f i c a t i o n i s shown i n F i g u r e 1 0 . 1 ( b ) . Comparing (a) and (b) t h e c u r v a t u r e of t h e i s o t h e r m s n e a r t h e i n t e r f a c e has i n c r e a s e d g i v i n g h i g h e r r a d i a l g r a d i e n t s . In (b) the a x i a l g r a d i e n t a t t h e i n t e r f a c e i s l e s s t h a n i n (a) and t h e d i s t a n c e above th e i n t e r f a c e where t h e a x i a l g r a d i e n t changes s i g n has i n c r e a s e d .  260 c d F i g u r e 10.1 T e m p e r a t u r e ( 1 0 3 o C ) f i e l d d u r i n g c o o l i n g a t f o u r t i m e s . (a) 5 s, (b) 10 s, (c) 20 s f a (d) 60 s, I n i t i a l g r a d i e n t c l o s e t o i n t e r f a c e 70 C/cm. A r g o n t e m p e r a t u r e 1000 C 261 The t e m p e r a t u r e d i s t r i b u t i o n a f t e r 20 s e c o n d s ( F i g u r e 1 0 . 1 ( c ) ) shows c o n t i n u i n g h i g h r a d i a l g r a d i e n t s c l o s e t o t h e i n t e r f a c e and a c o n s i d e r a b l y d e c r e a s i n g a x i a l g r a d i e n t . A f t e r 60 s e c o n d s ( F i g u r e 1 0 . 1 ( d ) ) the t e m p e r a t u r e d i s t r i b u t i o n has become much more u n i f o r m . By 300 s e c o n d s , t h e c a l c u l a t i o n s show the t e m p e r a t u r e d i s t r i b u t i o n i s a l m o s t u n i f o r m t h r o u g h o u t t h e c r y s t a l . The v e r y r a p i d c o o l i n g i n t h e f i r s t f i v e m i n u t e s a f t e r s o l i d i f i c a t i o n s u p p o r t s the q u a s i - s t e a d y s t a t e a p p r o x i m a t i o n assumed i n t h e c a l c u l a t i o n s o f t h e t e m p e r a t u r e d i s t r i b u t i o n d u r i n g c r y s t a l g r o w t h . F u r t h e r e x a m i n a t i o n of t h e i s o t h e r m s i n F i g u r e 1 0 . 1 ( a - d ) shows t h a t mixed c o n t r o l of t h e h e a t t r a n s f e r p r o c e s s o c c u r s d u r i n g c o o l i n g . T h i s can be c o n c l u d e d from t h e o b s e r v a t i o n t h a t the 1000°C i s o t h e r m does not move c o n s i d e r a b l y from a t o d i n F i g u r e 10.1. Heat t r a n s f e r c o n t r o l a t t h e s u r f a c e would c o n s i d e r a b l y move su c h i s o t h e r m upwards i n t h e c r y s t a l . On t h e o t h e r hand, h e a t t r a n s f e r c o n t r o l i n t h e b u l k c r y s t a l would g i v e l a r g e r t h a n t h e o b s e r v e d r a d i a l g r a d i e n t . The MRSS-CRSS ( Y i e l d ) d e t e r m i n e d from the t e m p e r a t u r e d i s t r i b u t i o n s shown i n F i g u r e l O . l ( a - d ) a r e shown i n F i g u r e 1 0 . 2 ( a - d ) . The l a r g e s t s t r e s s e s a r e o b s e r v e d t o o c c u r a t t h e i n t e r f a c e . A low s t r e s s r e g i o n i s p r e s e n t a b o u t 1 cm above t h e i n t e r f a c e midway between the o u t s i d e s u r f a c e and the v e r t i c a l a x i s w h i c h moves towa r d s t h e a x i s w i t h t i m e . As a r e s u l t of t h e  263 c d F i g u r e 10.2 MRSS-CRSS ( Y i e l d ) (MPa) d u r i n g c o o l i n g f o r the f o u r t e m p e r a t u r e f i e l d s shown i n F i g u r e l O . l ( a - d ) . (a) 5 s, (b) 10 s, ( c ) 20 s, (d) 60 s. I n i t i a l g r a d i e n t 70°C/cm. A r g o n t e m p e r a t u r e 1000 C. 264 movement the s t r e s s d i s t r i b u t i o n a l o n g the c r y s t a l d i a m e t e r changes from a W t o a U shape w i t h t i m e . The l a r g e s t MRSS-CRSS ( Y i e l d ) near t h e i n t e r f a c e o c c u r s about 10 s e c o n d s a f t e r g r o w t h s t o p s , and r a n g e s between 18 and 4.5 MPa. These s t r e s s v a l u e s a r e a p p r e c i a b l y l a r g e r t h a n the v a l u e s at t h e same p o s i t i o n i n t h e c r y s t a l d u r i n g growth and a r e abo u t 3 t i m e s t h e s t r e s s i n t h e c r y s t a l above t h e b o r o n o x i d e when growth i s s t o p p e d . T h i s would r e s u l t i n a t h r e e - f o l d i n c r e a s e i n d i s l o c a t i o n d e n s i t y i n the v i c i n i t y of t h e end o f the c r y s t a l g e n e r a t e d d u r i n g c o o l i n g . The MRSS-CRSS ( Y i e l d ) f o r c r y s t a l c r o s s - s e c t i o n s a t 2.75, 8.25, 16.5 and 27.5 mm from t h e i n t e r f a c e a r e shown i n F i g u r e 1 0 . 3 ( a - d ) , the main c h a r a c t e r i s t i c s of the s t r e s s s y m m e t r i e s o b s e r v e d a r e : 1) At 2.75 mm from t h e i n t e r f a c e t h e o u t e r r i n g has an e i g h t - f o l d symmetry and the s t r e s s d i s t r i b u t i o n a c r o s s a d i a m e t e r i s U s h a p e d . 2) At 8.25 mm from the i n t e r f a c e the c e n t r e has f o u r - f o l d symmetry and t w o - f o l d symmetry.. 3) At 16.5 mm from t h e i n t e r f a c e t h e r e i s c i r c u l a r symmetry. T h e r e a r e f o u r minimum s t r e s s r e g i o n s i n t h e <100> d i r e c t i o n s . 4) At 27.5 mm from t h e i n t e r f a c e t h e r e i s t w o - f o l d symmetry. 265 2 6 6 267 C 268 F i g u r e 10.3 MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n (001) p l a n e s a t f o u r d i s t a n c e s from t h e bottom ( i n t e r f a c e ) i n t h e c r y s t a l shown i n F i g u r e 10.2(b) 10 s, (a) 2.75 mm. (b) 8.25 mm ( c ) 16.5 mm (d) 27.5 mm . 269 The s t r e s s d i s t r i b u t i o n and symmetry a r e t i m e d e p e n d e n t . At 2.75 mm from t h e i n t e r f a c e t h e e i g h t - f o l d symmetry i n F i g u r e 1 0 . 3(a) a t 10 s e c o n d s changes at 60 s e c o n d s to the e i g h t - f o l d symmetry shown i n F i g u r e 1 0 . 4 ( a ) . At 8.25 mm f r o m th e i n t e r f a c e , t h e two and f o u r - f o l d s y m m e t r i e s i n F i g u r e 10.3(b) a t 10 s e c o n d s change at 60 s e c o n d s t o the t w o - f o l d symmetry shown i n F i g u r e 1 0 . 4 ( b ) . 10.1.1.2 I n i t i a l G r a d i e n t GC/2 = 35°C/cm In t h i s c a s e the s t r e s s e s r e a c h a maximum a t 10 s e c o n d s . The t e m p e r a t u r e f i e l d i s shown i n F i g u r e 1 0 . 5 ( a ) and t h e c o r r e s p o n d i n g MRSS-CRSS ( Y i e l d ) f i e l d i n F i g u r e 1 0 . 5 ( b ) . C o m p a r i n g F i g u r e 10.5(a) w i t h F i g u r e 1 0 .4(a) f o r a 70°C/cm g r a d i e n t i t i s n o t e d t h a t the i s o t h e r m s f o r t h e 35°C/cm g r a d i e n t have l a r g e r c u r v a t u r e t h a n f o r t h e 70°C/cm g r a d i e n t . The MRSS- CRSS ( Y i e l d ) d i s t r i b u t i o n i n F i g u r e 10.5(b) i s s i m i l a r t o t h e d i s t r i b u t i o n i n F i g u r e 10.4(b) f o r t h e 70°C/cm g r a d i e n t . L a r g e c u r v a t u r e i n t h e i s o t h e r m s and l a r g e s t r e s s e s a r e s t i l l p r e s e n t at 60 s e c o n d s as shown i n F i g u r e 1 0 . 6 ( a - b ) . The MRSS-CRSS ( Y i e l d ) i n a c r o s s - s e c t i o n a t 2.75 mm from th e i n t e r f a c e has e i g h t - f o l d symmetry. At 8.25 mm t h e s t r e s s has a s t r o n g t w o - f o l d symmetry ( F i g u r e 1 0 . 7 ( b ) ) . The t w o - f o l d symmetry g r a d u a l l y changes t o a n e a r c i r c u l a r symmetry w i t h s m a l l t r a c e s o f f o u r - f o l d symmetry g i v e n by weak l o c a l minima i n t h e <110> d i r e c t i o n s as shown i n F i g u r e 1 0 . 7 ( b ) . The U-shaped s t r e s s d i s t r i b u t i o n s a l o n g a d i a m e t e r i s p r e s e n t i n a l l c r o s s - s e c t i o n s . 270 271 F i g u r e 10.4 MRSS-CRSS ( Y i e l d ) (MPa) i n (001) p l a n e s at two d i s t a n c e s from t h e bottom i n t h e c r y s t a l shown i n F i g u r e 10.2 (d) 60 s, (a) 2.75 mm (b) 8.25 mm. 272 b F i g u r e 10.5 (a) T e m p e r a t u r e f i e l d (10 C) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n at 1000 C a f t e r 10 s. I n i t i a l g r a d i e n t 35°C/cm. 273 F i g u r e 10.6 (a) T e m p e r a t u r e f i e l d (10°°C) and (b) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 1000 C a f t e r 60 s. I n i t i a l g r a d i e n t 3 5°C/cm. 274 a 275 F i g u r e 10.7 MRSS-CRSS ( Y i e l d ) (MPa) i n (001) p l a n e s a t two d i s t a n c e s from t h e bottom i n t h e c r y s t a l shown i n F i g u r e 10.5 (b) 10 s. (a) 8.25 mm (b) 27.5 mm. 276 10.1.1.3 I n i t i a l G r a d i e n t GC/4 = 17.5°c/cm The t e m p e r a t u r e f i e l d ( F i g u r e 1 0 . 8 ( a ) ) g i v e s i s o t h e r m s w h i c h have t h e h i g h e s t c u r v a t u r e among t h e i n i t i a l g r a d i e n t s c o n s i d e r e d . The MRSS-CRSS ( Y i e l d ) ( F i g u r e 10.8(b) f i e l d i n t h e r e g i o n c l o s e to the i n t e r f a c e i s s i m i l a r to the p r e v i o u s c a s e s . The r e g i o n w i t h l a r g e s t r e s s n e a r t h e c r y s t a l s u r f a c e and a d j a c e n t t o t h e i n t e r f a c e i s expanded t o encompass n e a r l y a l l o f th e c r y s t a l . The s t r e s s d i s t r i b u t i o n a l o n g t h e d i a m e t e r i s U- sh a p e d . The MRSS-CRSS ( Y i e l d ) i n c r o s s - s e c t i o n s shows 1) e i g h t - f o l d symmetry c l o s e t o t h e i n t e r f a c e 2) t w o - f o l d symmetry above t h e i n t e r f a c e and up t o a d i s t a n c e from the i n t e r f a c e of 16.5 mm 3) Near c i r c u l a r symmetry a t 27.5 mm from t h e i n t e r f a c e w i t h low s t r e s s e s i n t h e c e n t r e c o v e r i n g more t h a n h a l f t h e s e c t i o n s u r f a c e a r e a and l a r g e s t r e s s g r a d i e n t s a t the edge. These g i v e a U-shaped s t r e s s d i s t r i b u t i o n w i t h a b r o a d b a s e . The l a r g e s t r e s s e s p e r s i s t a f t e r 60 s e c o n d s but t h e y a r e not l a r g e r t h a n t h e ones o b s e r v e d a t 10 s e c o n d s . The s t r e s s d i s t r i b u t i o n i n c r o s s - s e c t i o n s a t 60 s e c o n d s i s s i m i l a r t o t h e d i s t r i b u t i o n a t 10 s e c o n d s . A l o n g a d i a m e t e r t h e s t r e s s d i s t r i b u t i o n t e n d s t o t a k e a V shape r a t h e r t h a n U shape. At a d i s t a n c e o f 27.5 mm from the i n t e r f a c e t h e c e n t r a l r e g i o n i n t h e 277 F i g u r e 10.8 (a) T e m p e r a t u r e ( Y i e l d ) (MPa) at 1000 C a f t e r f i e l d ( 1 0 O U C ) and (b) MRSS-CRSS f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n 10 s. I n i t i a l g r a d i e n t 17.5°C/cm. 278 w a f e r shows a w e l l d i s t i n g u i s h e d e i g h t - f o l d symmetry ( F i g u r e 1 0 . 9 ) . The s t r e s s l e v e l i n t h i s s e c t i o n i s h i g h e r t h a n a t 10 s e c o n d s and t h e r e f o r e the symmetry may be e v i d e n t i n t h e d i s l o c a t i o n d e n s i t y d i s t r i b u t i o n . 10.1.2. Boron O x i d e The c a l c u l a t i o n s a r e r e p e a t e d f o r t h e same i n i t i a l t h e r m a l f i e l d s i n t h e c r y s t a l used above 70°C/cm and 17.5°C/cm and a s s u m i n g the s u r r o u n d i n g medium i s b o r o n o x i d e . The t e m p e r a t u r e and s t r e s s f i e l d s f o r t h e t h r e e t h e r m a l g r a d i e n t s GC, GC/2 and GC/4 a r e shown i n F i g u r e 10.10 t o 10.12. P a r t s a and b show th e two f i e l d s i n t h a t o r d e r . R e s u l t s a t 10 s e c o n d s a r e shown which e x h i b i t s the h i g h e s t c o o l i n g s t r e s s e s . The MRSS-CRSS ( Y i e l d ) d i s t r i b u t i o n s i n F i g u r e 10.10(b) t o 10.12(b) a r e s i m i l a r i n shape t o the d i s t r i b u t i o n s o b t a i n e d w i t h an a r g o n a t m o s p h e r e , but s t r e s s e s a r e 20 * h i g h e r i n m a g n i t u d e .